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<body>
  <div class="header">
    <h1 class="chapter_number">
      <a href="">CHAPTER 6</a>
    </h1>
    <h1 class="chapter_title"><a href="">Deep learning</a></h1>
  </div>
  <div class="section">
    <div id="toc">
      <p class="toc_title"><a href="index.html">Neural Networks and Deep Learning</a></p>
      <p class="toc_not_mainchapter"><a href="about.html">What this book is about</a></p>
      <p class="toc_not_mainchapter"><a href="exercises_and_problems.html">On the exercises and problems</a></p>
      <p class='toc_mainchapter'><a id="toc_using_neural_nets_to_recognize_handwritten_digits_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_using_neural_nets_to_recognize_handwritten_digits" src="images/arrow.png" width="15px"></a><a
          href="chap1.html">Using neural nets to recognize handwritten digits</a>
      <div id="toc_using_neural_nets_to_recognize_handwritten_digits" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap1.html#perceptrons">
            <li>Perceptrons</li>
          </a><a href="chap1.html#sigmoid_neurons">
            <li>Sigmoid neurons</li>
          </a><a href="chap1.html#the_architecture_of_neural_networks">
            <li>The architecture of neural networks</li>
          </a><a href="chap1.html#a_simple_network_to_classify_handwritten_digits">
            <li>A simple network to classify handwritten digits</li>
          </a><a href="chap1.html#learning_with_gradient_descent">
            <li>Learning with gradient descent</li>
          </a><a href="chap1.html#implementing_our_network_to_classify_digits">
            <li>Implementing our network to classify digits</li>
          </a><a href="chap1.html#toward_deep_learning">
            <li>Toward deep learning</li>
          </a></ul>
        </p>
      </div>
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      <p class='toc_mainchapter'><a id="toc_how_the_backpropagation_algorithm_works_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_how_the_backpropagation_algorithm_works" src="images/arrow.png" width="15px"></a><a
          href="chap2.html">How the backpropagation algorithm works</a>
      <div id="toc_how_the_backpropagation_algorithm_works" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap2.html#warm_up_a_fast_matrix-based_approach_to_computing_the_output
_from_a_neural_network">
            <li>Warm up: a fast matrix-based approach to computing the output
              from a neural network</li>
          </a><a href="chap2.html#the_two_assumptions_we_need_about_the_cost_function">
            <li>The two assumptions we need about the cost function</li>
          </a><a href="chap2.html#the_hadamard_product_$s_\odot_t$">
            <li>The Hadamard product, $s \odot t$</li>
          </a><a href="chap2.html#the_four_fundamental_equations_behind_backpropagation">
            <li>The four fundamental equations behind backpropagation</li>
          </a><a href="chap2.html#proof_of_the_four_fundamental_equations_(optional)">
            <li>Proof of the four fundamental equations (optional)</li>
          </a><a href="chap2.html#the_backpropagation_algorithm">
            <li>The backpropagation algorithm</li>
          </a><a href="chap2.html#the_code_for_backpropagation">
            <li>The code for backpropagation</li>
          </a><a href="chap2.html#in_what_sense_is_backpropagation_a_fast_algorithm">
            <li>In what sense is backpropagation a fast algorithm?</li>
          </a><a href="chap2.html#backpropagation_the_big_picture">
            <li>Backpropagation: the big picture</li>
          </a></ul>
        </p>
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      <p class='toc_mainchapter'><a id="toc_improving_the_way_neural_networks_learn_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_improving_the_way_neural_networks_learn" src="images/arrow.png" width="15px"></a><a
          href="chap3.html">Improving the way neural networks learn</a>
      <div id="toc_improving_the_way_neural_networks_learn" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap3.html#the_cross-entropy_cost_function">
            <li>The cross-entropy cost function</li>
          </a><a href="chap3.html#overfitting_and_regularization">
            <li>Overfitting and regularization</li>
          </a><a href="chap3.html#weight_initialization">
            <li>Weight initialization</li>
          </a><a href="chap3.html#handwriting_recognition_revisited_the_code">
            <li>Handwriting recognition revisited: the code</li>
          </a><a href="chap3.html#how_to_choose_a_neural_network's_hyper-parameters">
            <li>How to choose a neural network's hyper-parameters?</li>
          </a><a href="chap3.html#other_techniques">
            <li>Other techniques</li>
          </a></ul>
        </p>
      </div>
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      <p class='toc_mainchapter'><a id="toc_a_visual_proof_that_neural_nets_can_compute_any_function_reveal"
          class="toc_reveal" onMouseOver="this.style.borderBottom='1px solid #2A6EA6';"
          onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_a_visual_proof_that_neural_nets_can_compute_any_function" src="images/arrow.png"
            width="15px"></a><a href="chap4.html">A visual proof that neural nets can compute any function</a>
      <div id="toc_a_visual_proof_that_neural_nets_can_compute_any_function" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap4.html#two_caveats">
            <li>Two caveats</li>
          </a><a href="chap4.html#universality_with_one_input_and_one_output">
            <li>Universality with one input and one output</li>
          </a><a href="chap4.html#many_input_variables">
            <li>Many input variables</li>
          </a><a href="chap4.html#extension_beyond_sigmoid_neurons">
            <li>Extension beyond sigmoid neurons</li>
          </a><a href="chap4.html#fixing_up_the_step_functions">
            <li>Fixing up the step functions</li>
          </a><a href="chap4.html#conclusion">
            <li>Conclusion</li>
          </a></ul>
        </p>
      </div>
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      <p class='toc_mainchapter'><a id="toc_why_are_deep_neural_networks_hard_to_train_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_why_are_deep_neural_networks_hard_to_train" src="images/arrow.png" width="15px"></a><a
          href="chap5.html">Why are deep neural networks hard to train?</a>
      <div id="toc_why_are_deep_neural_networks_hard_to_train" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap5.html#the_vanishing_gradient_problem">
            <li>The vanishing gradient problem</li>
          </a><a href="chap5.html#what's_causing_the_vanishing_gradient_problem_unstable_gradients_in_deep_neural_nets">
            <li>What's causing the vanishing gradient problem? Unstable gradients in deep neural nets</li>
          </a><a href="chap5.html#unstable_gradients_in_more_complex_networks">
            <li>Unstable gradients in more complex networks</li>
          </a><a href="chap5.html#other_obstacles_to_deep_learning">
            <li>Other obstacles to deep learning</li>
          </a></ul>
        </p>
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      <p class='toc_mainchapter'><a id="toc_deep_learning_reveal" class="toc_reveal"
          onMouseOver="this.style.borderBottom='1px solid #2A6EA6';" onMouseOut="this.style.borderBottom='0px';"><img
            id="toc_img_deep_learning" src="images/arrow.png" width="15px"></a><a href="chap6.html">Deep learning</a>
      <div id="toc_deep_learning" style="display: none;">
        <p class="toc_section">
        <ul><a href="chap6.html#introducing_convolutional_networks">
            <li>Introducing convolutional networks</li>
          </a><a href="chap6.html#convolutional_neural_networks_in_practice">
            <li>Convolutional neural networks in practice</li>
          </a><a href="chap6.html#the_code_for_our_convolutional_networks">
            <li>The code for our convolutional networks</li>
          </a><a href="chap6.html#recent_progress_in_image_recognition">
            <li>Recent progress in image recognition</li>
          </a><a href="chap6.html#other_approaches_to_deep_neural_nets">
            <li>Other approaches to deep neural nets</li>
          </a><a href="chap6.html#on_the_future_of_neural_networks">
            <li>On the future of neural networks</li>
          </a></ul>
        </p>
      </div>
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      <p class="toc_not_mainchapter"><a href="sai.html">Appendix: Is there a <em>simple</em> algorithm for
          intelligence?</a></p>
      <p class="toc_not_mainchapter"><a href="acknowledgements.html">Acknowledgements</a></p>
      <p class="toc_not_mainchapter"><a href="faq.html">Frequently Asked Questions</a></p>
      <hr>
      <span class="sidebar_title">Resources</span>

      <p class="sidebar"><a href="https://twitter.com/michael_nielsen">Michael Nielsen on Twitter</a></p>

      <p class="sidebar"><a href="faq.html">Book FAQ</a></p>

      <p class="sidebar">
        <a href="https://github.com/mnielsen/neural-networks-and-deep-learning">Code repository</a>
      </p>

      <p class="sidebar">
        <a href="http://eepurl.com/0Xxjb">Michael Nielsen's project announcement mailing list</a>
      </p>

      <p class="sidebar"> <a href="http://www.deeplearningbook.org/">Deep Learning</a>, book by Ian
        Goodfellow, Yoshua Bengio, and Aaron Courville</p>

      <p class="sidebar"><a href="http://cognitivemedium.com">cognitivemedium.com</a></p>

      <hr>
      <a href="http://michaelnielsen.org"><img src="assets/Michael_Nielsen_Web_Small.jpg" width="160px"
          style="border-style: none;" /></a>

      <p class="sidebar">
        By <a href="http://michaelnielsen.org">Michael Nielsen</a> / Dec 2019
      </p>
    </div>
    </p>
    <p>In the <a href="chap5.html">last chapter</a> we learned that deep neural
      networks are often much harder to train than shallow neural networks.
      That's unfortunate, since we have good reason to believe that
      <em>if</em> we could train deep nets they'd be much more powerful than
      shallow nets. But while the news from the last chapter is
      discouraging, we won't let it stop us. In this chapter, we'll develop
      techniques which can be used to train deep networks, and apply them in
      practice. We'll also look at the broader picture, briefly reviewing
      recent progress on using deep nets for image recognition, speech
      recognition, and other applications. And we'll take a brief,
      speculative look at what the future may hold for neural nets, and for
      artificial intelligence.
    </p>
    <p>The chapter is a long one. To help you navigate, let's take a tour.
      The sections are only loosely coupled, so provided you have some basic
      familiarity with neural nets, you can jump to whatever most interests
      you.</p>
    <p>The <a href="#convolutional_networks">main part of the chapter</a> is an
      introduction to one of the most widely used types of deep network:
      deep convolutional networks. We'll work through a detailed example
      - code and all - of using convolutional nets to solve the problem
      of classifying handwritten digits from the MNIST data set:</p>
    <p>
      <center><img src="images/digits.png" width="160px"></center>
    </p>
    <p>We'll start our account of convolutional networks with the shallow
      networks used to attack this problem earlier in the book. Through
      many iterations we'll build up more and more powerful networks. As we
      go we'll explore many powerful techniques: convolutions, pooling, the
      use of GPUs to do far more training than we did with our shallow
      networks, the algorithmic expansion of our training data (to reduce
      overfitting), the use of the dropout technique (also to reduce
      overfitting), the use of ensembles of networks, and others. The
      result will be a system that offers near-human performance. Of the
      10,000 MNIST test images - images not seen during training! - our
      system will classify 9,967 correctly. Here's a peek at the 33 images
      which are misclassified. Note that the correct classification is in
      the top right; our program's classification is in the bottom right:</p>
    <p>
      <center><img src="images/ensemble_errors.png" width="580px"></center>
    </p>
    <p>Many of these are tough even for a human to classify. Consider, for
      example, the third image in the top row. To me it looks more like a
      "9" than an "8", which is the official classification. Our
      network also thinks it's a "9". This kind of "error" is at the
      very least understandable, and perhaps even commendable. We conclude
      our discussion of image recognition with a
      <a href="#recent_progress_in_image_recognition">survey of some of the
        spectacular recent progress</a> using networks (particularly
      convolutional nets) to do image recognition.
    </p>
    <p>The remainder of the chapter discusses deep learning from a broader
      and less detailed perspective. We'll
      <a href="#things_we_didn't_cover_but_which_you'll_eventually_want_to_know">briefly
        survey other models of neural networks</a>, such as recurrent neural
      nets and long short-term memory units, and how such models can be
      applied to problems in speech recognition, natural language
      processing, and other areas. And we'll
      <a href="#on_the_future_of_neural_networks">speculate about the
        future of neural networks and deep learning</a>, ranging from ideas
      like intention-driven user interfaces, to the role of deep learning in
      artificial intelligence.
    </p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>The chapter builds on the earlier chapters in the book, making use of
      and integrating ideas such as backpropagation, regularization, the
      softmax function, and so on. However, to read the chapter you don't
      need to have worked in detail through all the earlier chapters. It
      will, however, help to have read <a href="chap1.html">Chapter 1</a>, on the
      basics of neural networks. When I use concepts from Chapters 2 to 5,
      I provide links so you can familiarize yourself, if necessary.</p>
    <p>It's worth noting what the chapter is not. It's not a tutorial on the
      latest and greatest neural networks libraries. Nor are we going to be
      training deep networks with dozens of layers to solve problems at the
      very leading edge. Rather, the focus is on understanding some of the
      core principles behind deep neural networks, and applying them in the
      simple, easy-to-understand context of the MNIST problem. Put another
      way: the chapter is not going to bring you right up to the frontier.
      Rather, the intent of this and earlier chapters is to focus on
      fundamentals, and so to prepare you to understand a wide range of
      current work.</p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>
    <h3><a name="introducing_convolutional_networks"></a><a href="#introducing_convolutional_networks">Introducing
        convolutional networks</a></h3>
    </p>
    <p>
      In earlier chapters, we taught our neural networks to do a pretty good
      job recognizing images of handwritten digits:</p>
    <p>
      <center><img src="images/digits.png" width="160px"></center>
    </p>
    <p>We did this using networks in which adjacent network layers are fully
      connected to one another. That is, every neuron in the network is
      connected to every neuron in adjacent layers:</p>
    <p>
      <center>
        <img src="images/tikz41.png" />
      </center>
    </p>
    <p>In particular, for each pixel in the input image, we encoded the
      pixel's intensity as the value for a corresponding neuron in the input
      layer. For the $28 \times 28$ pixel images we've been using, this
      means our network has $784$ ($= 28 \times 28$) input neurons. We then
      trained the network's weights and biases so that the network's output
      would - we hope! - correctly identify the input image: '0', '1',
      '2', ..., '8', or '9'.</p>
    <p>Our earlier networks work pretty well: we've
      <a href="chap3.html#98percent">obtained a classification accuracy better
        than 98 percent</a>, using training and test data from the
      <a href="chap1.html#learning_with_gradient_descent">MNIST handwritten
        digit data set</a>. But upon reflection, it's strange to use networks
      with fully-connected layers to classify images. The reason is that
      such a network architecture does not take into account the spatial
      structure of the images. For instance, it treats input pixels which
      are far apart and close together on exactly the same footing. Such
      concepts of spatial structure must instead be inferred from the
      training data. But what if, instead of starting with a network
      architecture which is <em>tabula rasa</em>, we used an architecture
      which tries to take advantage of the spatial structure? In this
      section I describe <em>convolutional neural networks</em>*<span class="marginnote">
        *The
        origins of convolutional neural networks go back to the 1970s. But
        the seminal paper establishing the modern subject of convolutional
        networks was a 1998 paper,
        <a href="http://yann.lecun.com/exdb/publis/pdf/lecun-98.pdf">"Gradient-based
          learning applied to document recognition"</a>, by Yann LeCun,
        Léon Bottou, Yoshua Bengio, and Patrick Haffner.
        LeCun has since made an interesting
        <a href="https://www.facebook.com/yann.lecun/posts/10152348155137143">remark</a>
        on the terminology for convolutional nets: "The [biological] neural
        inspiration in models like convolutional nets is very
        tenuous. That's why I call them 'convolutional nets' not
        'convolutional neural nets', and why we call the nodes 'units' and
        not 'neurons' ". Despite this remark, convolutional nets use many
        of the same ideas as the neural networks we've studied up to now:
        ideas such as backpropagation, gradient descent, regularization,
        non-linear activation functions, and so on. And so we will follow
        common practice, and consider them a type of neural network. I will
        use the terms "convolutional neural network" and "convolutional
        net(work)" interchangeably. I will also use the terms
        "[artificial] neuron" and "unit" interchangeably.</span>. These
      networks use a special architecture which is particularly well-adapted
      to classify images. Using this architecture makes convolutional
      networks fast to train. This, in turn, helps us train deep,
      many-layer networks, which are very good at classifying images.
      Today, deep convolutional networks or some close variant are used in
      most neural networks for image recognition.
    </p>
    <p>Convolutional neural networks use three basic ideas: <em>local
        receptive fields</em>, <em>shared weights</em>, and <em>pooling</em>. Let's
      look at each of these ideas in turn.</p>
    <p><strong>Local receptive fields:</strong> In the fully-connected layers shown
      earlier, the inputs were depicted as a vertical line of neurons. In a
      convolutional net, it'll help to think instead of the inputs as a $28
      \times 28$ square of neurons, whose values correspond to the $28
      \times 28$ pixel intensities we're using as inputs:</p>
    <p>
      <center>
        <img src="images/tikz42.png" />
      </center>
    </p>
    <p>As per usual, we'll connect the input pixels to a layer of hidden
      neurons. But we won't connect every input pixel to every hidden
      neuron. Instead, we only make connections in small, localized regions
      of the input image.</p>
    <p>To be more precise, each neuron in the first hidden layer will be
      connected to a small region of the input neurons, say, for example, a
      $5 \times 5$ region, corresponding to $25$ input pixels. So, for a
      particular hidden neuron, we might have connections that look like
      this:
      <center>
        <img src="images/tikz43.png" />
      </center>
    </p>
    <p>That region in the input image is called the <em>local receptive
        field</em> for the hidden neuron. It's a little window on the input
      pixels. Each connection learns a weight. And the hidden neuron
      learns an overall bias as well. You can think of that particular
      hidden neuron as learning to analyze its particular local receptive
      field.</p>
    <p>We then slide the local receptive field across the entire input image.
      For each local receptive field, there is a different hidden neuron in
      the first hidden layer. To illustrate this concretely, let's start
      with a local receptive field in the top-left corner:
      <center>
        <img src="images/tikz44.png" />
      </center>
    </p>
    <p>Then we slide the local receptive field over by one pixel to the right
      (i.e., by one neuron), to connect to a second hidden neuron:</p>
    <p>
      <center>
        <img src="images/tikz45.png" />
      </center>
    </p>
    <p>And so on, building up the first hidden layer. Note that if we have a
      $28 \times 28$ input image, and $5 \times 5$ local receptive fields,
      then there will be $24 \times 24$ neurons in the hidden layer. This
      is because we can only move the local receptive field $23$ neurons
      across (or $23$ neurons down), before colliding with the right-hand
      side (or bottom) of the input image.</p>
    <p>I've shown the local receptive field being moved by one pixel at a
      time. In fact, sometimes a different <em>stride length</em> is used.
      For instance, we might move the local receptive field $2$ pixels to
      the right (or down), in which case we'd say a stride length of $2$ is
      used. In this chapter we'll mostly stick with stride length $1$, but
      it's worth knowing that people sometimes experiment with different
      stride lengths*<span class="marginnote">
        *As was done in earlier chapters, if we're
        interested in trying different stride lengths then we can use
        validation data to pick out the stride length which gives the best
        performance. For more details, see the
        <a href="chap3.html#how_to_choose_a_neural_network's_hyper-parameters">earlier
          discussion</a> of how to choose hyper-parameters in a neural network.
        The same approach may also be used to choose the size of the local
        receptive field - there is, of course, nothing special about using
        a $5 \times 5$ local receptive field. In general, larger local
        receptive fields tend to be helpful when the input images are
        significantly larger than the $28 \times 28$ pixel MNIST images.</span>.</p>
    <p><strong>Shared weights and biases:</strong> I've said that each hidden neuron
      has a bias and $5 \times 5$ weights connected to its local receptive
      field. What I did not yet mention is that we're going to use the
      <em>same</em> weights and bias for each of the $24 \times 24$ hidden
      neurons. In other words, for the $j, k$th hidden neuron, the output
      is:
      <a class="displaced_anchor" name="eqtn125"></a>\begin{eqnarray}
      \sigma\left(b + \sum_{l=0}^4 \sum_{m=0}^4 w_{l,m} a_{j+l, k+m} \right).
      \tag{125}\end{eqnarray}
      Here, $\sigma$ is the neural activation function - perhaps the
      <a href="chap1.html#sigmoid_neurons">sigmoid function</a> we used in
      earlier chapters. $b$ is the shared value for the bias. $w_{l,m}$ is
      a $5 \times 5$ array of shared weights. And, finally, we use $a_{x,
      y}$ to denote the input activation at position $x, y$.
    </p>
    <p>This means that all the neurons in the first hidden layer detect
      exactly the same feature*<span class="marginnote">
        *I haven't precisely defined the
        notion of a feature. Informally, think of the feature detected by a
        hidden neuron as the kind of input pattern that will cause the
        neuron to activate: it might be an edge in the image, for instance,
        or maybe some other type of shape. </span>, just at different locations in
      the input image. To see why this makes sense, suppose the weights and
      bias are such that the hidden neuron can pick out, say, a vertical
      edge in a particular local receptive field. That ability is also
      likely to be useful at other places in the image. And so it is useful
      to apply the same feature detector everywhere in the image. To put it
      in slightly more abstract terms, convolutional networks are well
      adapted to the translation invariance of images: move a picture of a
      cat (say) a little ways, and it's still an image of a cat*<span class="marginnote">
        *In
        fact, for the MNIST digit classification problem we've been
        studying, the images are centered and size-normalized. So MNIST has
        less translation invariance than images found "in the wild", so to
        speak. Still, features like edges and corners are likely to be
        useful across much of the input space. </span>.</p>
    <p>For this reason, we sometimes call the map from the input layer to the
      hidden layer a <em>feature map</em>. We call the weights defining the
      feature map the <em>shared weights</em>. And we call the bias defining
      the feature map in this way the <em>shared bias</em>. The shared
      weights and bias are often said to define a <em>kernel</em> or
      <em>filter</em>. In the literature, people sometimes use these terms in
      slightly different ways, and for that reason I'm not going to be more
      precise; rather, in a moment, we'll look at some concrete examples.
    </p>
    <p></p>
    <p>The network structure I've described so far can detect just a single
      kind of localized feature. To do image recognition we'll need more
      than one feature map. And so a complete convolutional layer consists
      of several different feature maps:</p>
    <p>
      <center>
        <img src="images/tikz46.png" />
      </center>
      In the example shown, there are $3$ feature maps. Each feature map is
      defined by a set of $5 \times 5$ shared weights, and a single shared
      bias. The result is that the network can detect $3$ different kinds
      of features, with each feature being detectable across the entire
      image.
    </p>
    <p></p>
    <p>I've shown just $3$ feature maps, to keep the diagram above simple.
      However, in practice convolutional networks may use more (and perhaps
      many more) feature maps. One of the early convolutional networks,
      LeNet-5, used $6$ feature maps, each associated to a $5 \times 5$
      local receptive field, to recognize MNIST digits. So the example
      illustrated above is actually pretty close to LeNet-5. In the
      examples we develop later in the chapter we'll use convolutional
      layers with $20$ and $40$ feature maps. Let's take a quick peek at
      some of the features which are learned*<span class="marginnote">
        *The feature maps
        illustrated come from the final convolutional network we train, see
        <a href="#final_conv">here</a>.</span>:</p>
    <p>
      <center><img src="images/net_full_layer_0.png" width="400px"></center>
    </p>
    <p>The $20$ images correspond to $20$ different feature maps (or filters,
      or kernels). Each map is represented as a $5 \times 5$ block image,
      corresponding to the $5 \times 5$ weights in the local receptive
      field. Whiter blocks mean a smaller (typically, more negative)
      weight, so the feature map responds less to corresponding input
      pixels. Darker blocks mean a larger weight, so the feature map
      responds more to the corresponding input pixels. Very roughly
      speaking, the images above show the type of features the convolutional
      layer responds to.</p>
    <p>So what can we conclude from these feature maps? It's clear there is
      spatial structure here beyond what we'd expect at random: many of the
      features have clear sub-regions of light and dark. That shows our
      network really is learning things related to the spatial structure.
      However, beyond that, it's difficult to see what these feature
      detectors are learning. Certainly, we're not learning (say) the
      <a href="http://en.wikipedia.org/wiki/Gabor_filter">Gabor filters</a> which
      have been used in many traditional approaches to image recognition.
      In fact, there's now a lot of work on better understanding the
      features learnt by convolutional networks. If you're interested in
      following up on that work, I suggest starting with the paper
      <a href="http://arxiv.org/abs/1311.2901">Visualizing and Understanding
        Convolutional Networks</a> by Matthew Zeiler and Rob Fergus (2013).
    </p>
    <p></p>
    <p>A big advantage of sharing weights and biases is that it greatly
      reduces the number of parameters involved in a convolutional network.
      For each feature map we need $25 = 5 \times 5$ shared weights, plus a
      single shared bias. So each feature map requires $26$ parameters. If
      we have $20$ feature maps that's a total of $20 \times 26 = 520$
      parameters defining the convolutional layer. By comparison, suppose
      we had a fully connected first layer, with $784 = 28 \times 28$ input
      neurons, and a relatively modest $30$ hidden neurons, as we used in
      many of the examples earlier in the book. That's a total of $784
      \times 30$ weights, plus an extra $30$ biases, for a total of $23,550$
      parameters. In other words, the fully-connected layer would have more
      than $40$ times as many parameters as the convolutional layer.</p>
    <p>Of course, we can't really do a direct comparison between the number
      of parameters, since the two models are different in essential ways.
      But, intuitively, it seems likely that the use of translation
      invariance by the convolutional layer will reduce the number of
      parameters it needs to get the same performance as the fully-connected
      model. That, in turn, will result in faster training for the
      convolutional model, and, ultimately, will help us build deep networks
      using convolutional layers.</p>
    <p></p>
    <p>Incidentally, the name <em>convolutional</em> comes from the fact that
      the operation in Equation <span id="margin_736057893021_reveal" class="equation_link">(125)</span><span
        id="margin_736057893021" class="marginequation" style="display: none;"><a href="chap6.html#eqtn125"
          style="padding-bottom: 5px;" onMouseOver="this.style.borderBottom='1px solid #2A6EA6';"
          onMouseOut="this.style.borderBottom='0px';">\begin{eqnarray}
          \sigma\left(b + \sum_{l=0}^4 \sum_{m=0}^4 w_{l,m} a_{j+l, k+m} \right) \nonumber\end{eqnarray}</a></span>
      <script>$('#margin_736057893021_reveal').click(function () { $('#margin_736057893021').toggle('slow', function () { }); });</script>
      is sometimes known as a
      <em>convolution</em>. A little more precisely, people sometimes write
      that equation as $a^1 = \sigma(b + w * a^0)$, where $a^1$ denotes the
      set of output activations from one feature map, $a^0$ is the set of
      input activations, and $*$ is called a convolution operation. We're
      not going to make any deep use of the mathematics of convolutions, so
      you don't need to worry too much about this connection. But it's
      worth at least knowing where the name comes from.
    </p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p><strong>Pooling layers:</strong> In addition to the convolutional layers just
      described, convolutional neural networks also contain <em>pooling
        layers</em>. Pooling layers are usually used immediately after
      convolutional layers. What the pooling layers do is simplify the
      information in the output from the convolutional layer.</p>
    <p></p>
    <p>In detail, a pooling layer takes each feature map*<span class="marginnote">
        *The
        nomenclature is being used loosely here. In particular, I'm using
        "feature map" to mean not the function computed by the
        convolutional layer, but rather the activation of the hidden neurons
        output from the layer. This kind of mild abuse of nomenclature is
        pretty common in the research literature.</span> output from the
      convolutional layer and prepares a condensed feature map. For
      instance, each unit in the pooling layer may summarize a region of
      (say) $2 \times 2$ neurons in the previous layer. As a concrete
      example, one common procedure for pooling is known as
      <em>max-pooling</em>. In max-pooling, a pooling unit simply outputs the
      maximum activation in the $2 \times 2$ input region, as illustrated in
      the following diagram:
    </p>
    <p>
      <center>
        <img src="images/tikz47.png" />
      </center>
    </p>
    <p>Note that since we have $24 \times 24$ neurons output from the
      convolutional layer, after pooling we have $12 \times 12$ neurons.</p>
    <p>As mentioned above, the convolutional layer usually involves more than
      a single feature map. We apply max-pooling to each feature map
      separately. So if there were three feature maps, the combined
      convolutional and max-pooling layers would look like:</p>
    <p>
      <center>
        <img src="images/tikz48.png" />
      </center>
    </p>
    <p>We can think of max-pooling as a way for the network to ask whether a
      given feature is found anywhere in a region of the image. It then
      throws away the exact positional information. The intuition is that
      once a feature has been found, its exact location isn't as important
      as its rough location relative to other features. A big benefit is
      that there are many fewer pooled features, and so this helps reduce
      the number of parameters needed in later layers.</p>
    <p></p>
    <p>Max-pooling isn't the only technique used for pooling. Another common
      approach is known as <em>L2 pooling</em>. Here, instead of taking the
      maximum activation of a $2 \times 2$ region of neurons, we take the
      square root of the sum of the squares of the activations in the $2
      \times 2$ region. While the details are different, the intuition is
      similar to max-pooling: L2 pooling is a way of condensing information
      from the convolutional layer. In practice, both techniques have been
      widely used. And sometimes people use other types of pooling
      operation. If you're really trying to optimize performance, you may
      use validation data to compare several different approaches to
      pooling, and choose the approach which works best. But we're not
      going to worry about that kind of detailed optimization.</p>
    <p></p>
    <p><strong>Putting it all together:</strong> We can now put all these ideas
      together to form a complete convolutional neural network. It's
      similar to the architecture we were just looking at, but has the
      addition of a layer of $10$ output neurons, corresponding to the $10$
      possible values for MNIST digits ('0', '1', '2', <em>etc</em>):</p>
    <p>
      <center>
        <img src="images/tikz49.png" />
      </center>
    </p>
    <p>The network begins with $28 \times 28$ input neurons, which are used
      to encode the pixel intensities for the MNIST image. This is then
      followed by a convolutional layer using a $5 \times 5$ local receptive
      field and $3$ feature maps. The result is a layer of $3 \times 24
      \times 24$ hidden feature neurons. The next step is a max-pooling
      layer, applied to $2 \times 2$ regions, across each of the $3$ feature
      maps. The result is a layer of $3 \times 12 \times 12$ hidden feature
      neurons.</p>
    <p>The final layer of connections in the network is a fully-connected
      layer. That is, this layer connects <em>every</em> neuron from the
      max-pooled layer to every one of the $10$ output neurons. This
      fully-connected architecture is the same as we used in earlier
      chapters. Note, however, that in the diagram above, I've used a
      single arrow, for simplicity, rather than showing all the connections.
      Of course, you can easily imagine the connections.</p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>This convolutional architecture is quite different to the
      architectures used in earlier chapters. But the overall picture is
      similar: a network made of many simple units, whose behaviors are
      determined by their weights and biases. And the overall goal is still
      the same: to use training data to train the network's weights and
      biases so that the network does a good job classifying input digits.</p>
    <p>In particular, just as earlier in the book, we will train our network
      using stochastic gradient descent and backpropagation. This mostly
      proceeds in exactly the same way as in earlier chapters. However, we
      do need to make a few modifications to the backpropagation procedure.
      The reason is that our earlier <a href="chap2.html">derivation of
        backpropagation</a> was for networks with fully-connected layers.
      Fortunately, it's straightforward to modify the derivation for
      convolutional and max-pooling layers. If you'd like to understand the
      details, then I invite you to work through the following problem. Be
      warned that the problem will take some time to work through, unless
      you've really internalized the <a href="chap2.html">earlier derivation of
        backpropagation</a> (in which case it's easy).</p>
    <p>
    <h4><a name="problem_366128"></a><a href="#problem_366128">Problem</a></h4>
    <ul>
      <li><strong>Backpropagation in a convolutional network</strong> The core equations
        of backpropagation in a network with fully-connected layers
        are <span id="margin_209324202488_reveal" class="equation_link">(BP1)</span><span id="margin_209324202488"
          class="marginequation" style="display: none;"><a href="chap2.html#eqtnBP1" style="padding-bottom: 5px;"
            onMouseOver="this.style.borderBottom='1px solid #2A6EA6';"
            onMouseOut="this.style.borderBottom='0px';">\begin{eqnarray}
            \delta^L_j = \frac{\partial C}{\partial a^L_j} \sigma'(z^L_j) \nonumber\end{eqnarray}</a></span>
        <script>$('#margin_209324202488_reveal').click(function () { $('#margin_209324202488').toggle('slow', function () { }); });</script>
        -<span id="margin_759010070359_reveal" class="equation_link">(BP4)</span><span id="margin_759010070359"
          class="marginequation" style="display: none;"><a href="chap2.html#eqtnBP4" style="padding-bottom: 5px;"
            onMouseOver="this.style.borderBottom='1px solid #2A6EA6';"
            onMouseOut="this.style.borderBottom='0px';">\begin{eqnarray}
            \frac{\partial C}{\partial w^l_{jk}} = a^{l-1}_k \delta^l_j \nonumber\end{eqnarray}</a></span>
        <script>$('#margin_759010070359_reveal').click(function () { $('#margin_759010070359').toggle('slow', function () { }); });</script>
        (<a href="chap2.html#backpropsummary">link</a>). Suppose we have a
        network containing a convolutional layer, a max-pooling layer, and a
        fully-connected output layer, as in the network discussed above.
        How are the equations of backpropagation modified?
    </ul>
    </p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>
    <h3><a name="convolutional_neural_networks_in_practice"></a><a
        href="#convolutional_neural_networks_in_practice">Convolutional neural networks in practice</a></h3>
    </p>
    <p>We've now seen the core ideas behind convolutional neural networks.
      Let's look at how they work in practice, by implementing some
      convolutional networks, and applying them to the MNIST digit
      classification problem. The program we'll use to do this is called
      <tt>network3.py</tt>, and it's an improved version of the programs
      <tt>network.py</tt> and <tt>network2.py</tt> developed in earlier
      chapters*<span class="marginnote">
        *Note also that <tt>network3.py</tt> incorporates ideas
        from the Theano library's documentation on convolutional neural nets
        (notably the implementation of
        <a href="http://deeplearning.net/tutorial/lenet.html">LeNet-5</a>), from
        Misha Denil's
        <a href="https://github.com/mdenil/dropout">implementation of dropout</a>,
        and from <a href="http://colah.github.io">Chris Olah</a>.</span>. If you wish
      to follow along, the code is available
      <a href="https://github.com/mnielsen/neural-networks-and-deep-learning/blob/master/src/network3.py">on
        GitHub</a>. Note that we'll work through the code for
      <tt>network3.py</tt> itself in the next section. In this section, we'll
      use <tt>network3.py</tt> as a library to build convolutional networks.
    </p>
    <p></p>
    <p>The programs <tt>network.py</tt> and <tt>network2.py</tt> were implemented
      using Python and the matrix library Numpy. Those programs worked from
      first principles, and got right down into the details of
      backpropagation, stochastic gradient descent, and so on. But now that
      we understand those details, for <tt>network3.py</tt> we're going to use
      a machine learning library known as
      <a href="http://deeplearning.net/software/theano/">Theano</a>*<span class="marginnote">
        *See
        <a href="http://www.iro.umontreal.ca/&#126;lisa/pointeurs/theano_scipy2010.pdf">Theano:
          A CPU and GPU Math Expression Compiler in Python</a>, by James
        Bergstra, Olivier Breuleux, Frederic Bastien, Pascal Lamblin, Ravzan
        Pascanu, Guillaume Desjardins, Joseph Turian, David Warde-Farley,
        and Yoshua Bengio (2010). Theano is also the basis for the popular
        <a href="http://deeplearning.net/software/pylearn2/">Pylearn2</a> and
        <a href="http://keras.io/">Keras</a> neural networks libraries. Other
        popular neural nets libraries at the time of this writing include
        <a href="http://caffe.berkeleyvision.org">Caffe</a> and
        <a href="http://torch.ch">Torch</a>. </span>. Using Theano makes it easy to
      implement backpropagation for convolutional neural networks, since it
      automatically computes all the mappings involved. Theano is also
      quite a bit faster than our earlier code (which was written to be easy
      to understand, not fast), and this makes it practical to train more
      complex networks. In particular, one great feature of Theano is that
      it can run code on either a CPU or, if available, a GPU. Running on a
      GPU provides a substantial speedup and, again, helps make it practical
      to train more complex networks.
    </p>
    <p></p>
    <p>If you wish to follow along, then you'll need to get Theano running on
      your system. To install Theano, follow the instructions at the
      project's <a href="http://deeplearning.net/software/theano/">homepage</a>.
      The examples which follow were run using Theano 0.6*<span class="marginnote">
        *As I
        release this chapter, the current version of Theano has changed to
        version 0.7. I've actually rerun the examples under Theano 0.7 and
        get extremely similar results to those reported in the text.</span>. Some
      were run under Mac OS X Yosemite, with no GPU. Some were run on
      Ubuntu 14.04, with an NVIDIA GPU. And some of the experiments were run
      under both. To get <tt>network3.py</tt> running you'll need to set the
      <tt>GPU</tt> flag to either <tt>True</tt> or <tt>False</tt> (as appropriate)
      in the <tt>network3.py</tt> source. Beyond that, to get Theano up and
      running on a GPU you may find
      <a href="http://deeplearning.net/software/theano/tutorial/using_gpu.html">the
        instructions here</a> helpful. There are also tutorials on the web,
      easily found using Google, which can help you get things working. If
      you don't have a GPU available locally, then you may wish to look into
      <a href="http://aws.amazon.com/ec2/instance-types/">Amazon Web Services</a>
      EC2 G2 spot instances. Note that even with a GPU the code will take
      some time to execute. Many of the experiments take from minutes to
      hours to run. On a CPU it may take days to run the most complex of
      the experiments. As in earlier chapters, I suggest setting things
      running, and continuing to read, occasionally coming back to check the
      output from the code. If you're using a CPU, you may wish to reduce
      the number of training epochs for the more complex experiments, or
      perhaps omit them entirely.
    </p>
    <p>To get a baseline, we'll start with a shallow architecture using just
      a single hidden layer, containing $100$ hidden neurons. We'll train
      for $60$ epochs, using a learning rate of $\eta = 0.1$, a mini-batch
      size of $10$, and no regularization. Here we go*<span class="marginnote">
        *Code for the
        experiments in this section may be found
        <a href="https://github.com/mnielsen/neural-networks-and-deep-learning/blob/master/src/conv.py">in
          this script</a>. Note that the code in the script simply duplicates
        and parallels the discussion in this section.<br><br>Note also that
        throughout the section I've explicitly specified the number of
        training epochs. I've done this for clarity about how we're
        training. In practice, it's worth using
        <a href="chap3.html#early_stopping">early stopping</a>, that is,
        tracking accuracy on the validation set, and stopping training when
        we are confident the validation accuracy has stopped improving.</span>:</p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="kn">import</span> <span class="nn">network3</span>
<span class="o">&gt;&gt;&gt;</span> <span class="kn">from</span> <span class="nn">network3</span> <span class="kn">import</span> <span class="n">Network</span>
<span class="o">&gt;&gt;&gt;</span> <span class="kn">from</span> <span class="nn">network3</span> <span class="kn">import</span> <span class="n">ConvPoolLayer</span><span class="p">,</span> <span class="n">FullyConnectedLayer</span><span class="p">,</span> <span class="n">SoftmaxLayer</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">training_data</span><span class="p">,</span> <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span> <span class="o">=</span> <span class="n">network3</span><span class="o">.</span><span class="n">load_data_shared</span><span class="p">()</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">mini_batch_size</span> <span class="o">=</span> <span class="mi">10</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">784</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p></p>
    <p>
      I obtained a best classification accuracy of $97.80$ percent. This is
      the classification accuracy on the <tt>test_data</tt>, evaluated at the
      training epoch where we get the best classification accuracy on the
      <tt>validation_data</tt>. Using the validation data to decide when to
      evaluate the test accuracy helps avoid overfitting to the test data
      (see this <a href="chap3.html#validation_explanation">earlier
        discussion</a> of the use of validation data). We will follow this
      practice below. Your results may vary slightly, since the network's
      weights and biases are randomly initialized*<span class="marginnote">
        *In fact, in this
        experiment I actually did three separate runs training a network
        with this architecture. I then reported the test accuracy which
        corresponded to the best validation accuracy from any of the three
        runs. Using multiple runs helps reduce variation in results, which
        is useful when comparing many architectures, as we are doing. I've
        followed this procedure below, except where noted. In practice, it
        made little difference to the results obtained.</span>.
    </p>
    <p>This $97.80$ percent accuracy is close to the $98.04$ percent accuracy
      obtained back in <a href="chap3.html#chap3_98_04_percent">Chapter 3</a>,
      using a similar network architecture and learning hyper-parameters.
      In particular, both examples used a shallow network, with a single
      hidden layer containing $100$ hidden neurons. Both also trained for
      $60$ epochs, used a mini-batch size of $10$, and a learning rate of
      $\eta = 0.1$.</p>
    <p>There were, however, two differences in the earlier network. First,
      we <a href="chap3.html#overfitting_and_regularization">regularized</a>
      the earlier network, to help reduce the effects of
      overfitting. Regularizing the current network does improve the
      accuracies, but the gain is only small, and so we'll hold off worrying
      about regularization until later. Second, while the final layer in
      the earlier network used sigmoid activations and the cross-entropy
      cost function, the current network uses a softmax final layer, and the
      log-likelihood cost function. As
      <a href="chap3.html#softmax">explained</a> in Chapter 3 this isn't a big
      change. I haven't made this switch for any particularly deep reason
      - mostly, I've done it because softmax plus log-likelihood cost is
      more common in modern image classification networks.
    </p>
    <p>Can we do better than these results using a deeper network
      architecture?</p>
    <p>Let's begin by inserting a convolutional layer, right at the beginning
      of the network. We'll use $5$ by $5$ local receptive fields, a stride
      length of $1$, and $20$ feature maps. We'll also insert a max-pooling
      layer, which combines the features using $2$ by $2$ pooling windows.
      So the overall network architecture looks much like the architecture
      discussed in the last section, but with an extra fully-connected
      layer:</p>
    <p>
      <center><img src="images/simple_conv.png" width="550px"></center>
    </p>
    <p>In this architecture, we can think of the convolutional and pooling
      layers as learning about local spatial structure in the input training
      image, while the later, fully-connected layer learns at a more
      abstract level, integrating global information from across the entire
      image. This is a common pattern in convolutional neural networks.</p>
    <p>Let's train such a network, and see how it performs*<span class="marginnote">
        *I've
        continued to use a mini-batch size of $10$ here. In fact, as we
        <a href="chap3.html#mini_batch_size">discussed earlier</a> it may be
        possible to speed up training using larger mini-batches. I've
        continued to use the same mini-batch size mostly for consistency
        with the experiments in earlier chapters.</span>:</p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">20</span><span class="o">*</span><span class="mi">12</span><span class="o">*</span><span class="mi">12</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">)</span>   
</pre>
    </div>
    </p>
    <p></p>
    <p></p>
    <p>That gets us to $98.78$ percent accuracy, which is a considerable
      improvement over any of our previous results. Indeed, we've reduced
      our error rate by better than a third, which is a great improvement.</p>
    <p>In specifying the network structure, I've treated the convolutional
      and pooling layers as a single layer. Whether they're regarded as
      separate layers or as a single layer is to some extent a matter of
      taste. <tt>network3.py</tt> treats them as a single layer because it
      makes the code for <tt>network3.py</tt> a little more compact. However,
      it is easy to modify <tt>network3.py</tt> so the layers can be specified
      separately, if desired.</p>
    <p>
    <h4><a name="exercise_683491"></a><a href="#exercise_683491">Exercise</a></h4>
    <ul>
      <li> What classification accuracy do you get if you omit the
        fully-connected layer, and just use the convolutional-pooling layer
        and softmax layer? Does the inclusion of the fully-connected layer
        help?






    </ul>
    </p>
    <p>Can we improve on the $98.78$ percent classification accuracy?</p>
    <p>Let's try inserting a second convolutional-pooling layer. We'll make
      the insertion between the existing convolutional-pooling layer and the
      fully-connected hidden layer. Again, we'll use a $5 \times 5$ local
      receptive field, and pool over $2 \times 2$ regions. Let's see what
      happens when we train using similar hyper-parameters to before:</p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)),</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">40</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="mi">4</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">)</span>        
</pre>
    </div>
    </p>
    <p></p>
    <p></p>
    <p>Once again, we get an improvement: we're now at $99.06$ percent
      classification accuracy!</p>
    <p>There's two natural questions to ask at this point. The first
      question is: what does it even mean to apply a second
      convolutional-pooling layer? In fact, you can think of the second
      convolutional-pooling layer as having as input $12 \times 12$
      "images", whose "pixels" represent the presence (or absence) of
      particular localized features in the original input image. So you can
      think of this layer as having as input a version of the original input
      image. That version is abstracted and condensed, but still has a lot
      of spatial structure, and so it makes sense to use a second
      convolutional-pooling layer.</p>
    <p>That's a satisfying point of view, but gives rise to a second
      question. The output from the previous layer involves $20$ separate
      feature maps, and so there are $20 \times 12 \times 12$ inputs to the
      second convolutional-pooling layer. It's as though we've got $20$
      separate images input to the convolutional-pooling layer, not a single
      image, as was the case for the first convolutional-pooling layer. How
      should neurons in the second convolutional-pooling layer respond to
      these multiple input images? In fact, we'll allow each neuron in this
      layer to learn from <em>all</em> $20 \times 5 \times 5$ input neurons in
      its local receptive field. More informally: the feature detectors in
      the second convolutional-pooling layer have access to <em>all</em> the
      features from the previous layer, but only within their particular
      local receptive field*<span class="marginnote">
        *This issue would have arisen in the
        first layer if the input images were in color. In that case we'd
        have 3 input features for each pixel, corresponding to red, green
        and blue channels in the input image. So we'd allow the feature
        detectors to have access to all color information, but only within a
        given local receptive field.</span>.</p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>
    <h4><a name="problem_834310"></a><a href="#problem_834310">Problem</a></h4>
    <ul>
      <li><strong>Using the tanh activation function</strong> Several times earlier in the
        book I've mentioned arguments that the
        <a href="chap3.html#other_models_of_artificial_neuron">tanh
          function</a> may be a better activation function than the sigmoid
        function. We've never acted on those suggestions, since we were
        already making plenty of progress with the sigmoid. But now let's
        try some experiments with tanh as our activation function. Try
        training the network with tanh activations in the convolutional and
        fully-connected layers*<span class="marginnote">
          *Note that you can pass
          <tt>activation_fn=tanh</tt> as a parameter to the
          <tt>ConvPoolLayer</tt> and <tt>FullyConnectedLayer</tt> classes.</span>.
        Begin with the same hyper-parameters as for the sigmoid network, but
        train for $20$ epochs instead of $60$. How well does your network
        perform? What if you continue out to $60$ epochs? Try plotting the
        per-epoch validation accuracies for both tanh- and sigmoid-based
        networks, all the way out to $60$ epochs. If your results are
        similar to mine, you'll find the tanh networks train a little
        faster, but the final accuracies are very similar. Can you explain
        why the tanh network might train faster? Can you get a similar
        training speed with the sigmoid, perhaps by changing the learning
        rate, or doing some rescaling*<span class="marginnote">
          *You may perhaps find
          inspiration in recalling that $\sigma(z) = (1+\tanh(z/2))/2$.</span>?
        Try a half-dozen iterations on the learning hyper-parameters or
        network architecture, searching for ways that tanh may be superior
        to the sigmoid. <em>Note: This is an open-ended problem.
          Personally, I did not find much advantage in switching to tanh,
          although I haven't experimented exhaustively, and perhaps you may
          find a way. In any case, in a moment we will find an advantage in
          switching to the rectified linear activation function, and so we
          won't go any deeper into the use of tanh.</em>
    </ul>
    </p>
    <p></p>
    <p>
      <strong>Using rectified linear units:</strong> The network we've developed at
      this point is actually a variant of one of the networks used in the
      seminal 1998
      paper*<span class="marginnote">
        *<a href="http://yann.lecun.com/exdb/publis/pdf/lecun-98.pdf">"Gradient-based
          learning applied to document recognition"</a>, by Yann LeCun,
        Léon Bottou, Yoshua Bengio, and Patrick Haffner
        (1998). There are many differences of detail, but broadly speaking
        our network is quite similar to the networks described in the
        paper.</span> introducing the MNIST problem, a network known as LeNet-5.
      It's a good foundation for further experimentation, and for building
      up understanding and intuition. In particular, there are many ways we
      can vary the network in an attempt to improve our results.
    </p>
    <p>As a beginning, let's change our neurons so that instead of using a
      sigmoid activation function, we use
      <a href="chap3.html#other_models_of_artificial_neuron">rectified
        linear units</a>. That is, we'll use the activation function $f(z)
      \equiv \max(0, z)$. We'll train for $60$ epochs, with a learning rate
      of $\eta = 0.03$. I also found that it helps a little to use some
      <a href="chap3.html#overfitting_and_regularization">l2
        regularization</a>, with regularization parameter $\lambda = 0.1$:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="kn">from</span> <span class="nn">network3</span> <span class="kn">import</span> <span class="n">ReLU</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">40</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="mi">4</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">,</span> <span class="n">lmbda</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p></p>
    <p>I obtained a classification accuracy of $99.23$ percent. It's a
      modest improvement over the sigmoid results ($99.06$). However,
      across all my experiments I found that networks based on rectified
      linear units consistently outperformed networks based on sigmoid
      activation functions. There appears to be a real gain in moving to
      rectified linear units for this problem.</p>
    <p>What makes the rectified linear activation function better than the
      sigmoid or tanh functions? At present, we have a poor understanding
      of the answer to this question. Indeed, rectified linear units have
      only begun to be widely used in the past few years. The reason for
      that recent adoption is empirical: a few people tried rectified linear
      units, often on the basis of hunches or heuristic arguments*<span class="marginnote">
        *A
        common justification is that $\max(0, z)$ doesn't saturate in the
        limit of large $z$, unlike sigmoid neurons, and this helps rectified
        linear units continue learning. The argument is fine, as far it
        goes, but it's hardly a detailed justification, more of a just-so
        story. Note that we discussed the problems with saturation back in
        <a href="chap2.html#saturation">Chapter 2</a>.</span>. They got good
      results classifying benchmark data sets, and the practice has spread.
      In an ideal world we'd have a theory telling us which activation
      function to pick for which application. But at present we're a long
      way from such a world. I should not be at all surprised if further
      major improvements can be obtained by an even better choice of
      activation function. And I also expect that in coming decades a
      powerful theory of activation functions will be developed. Today, we
      still have to rely on poorly understood rules of thumb and experience.</p>
    <p><strong>Expanding the training data:</strong> Another way we may hope to
      improve our results is by algorithmically expanding the training data.
      A simple way of expanding the training data is to displace each
      training image by a single pixel, either up one pixel, down one pixel,
      left one pixel, or right one pixel. We can do this by running the
      program <tt>expand_mnist.py</tt> from the shell prompt*<span class="marginnote">
        *The code
        for <tt>expand_mnist.py</tt> is available
        <a
          href="https://github.com/mnielsen/neural-networks-and-deep-learning/blob/master/src/expand_mnist.py">here</a>.</span>:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span> 
$ python expand_mnist.py
</pre>
    </div>
    </p>
    <p>Running this program takes the $50,000$ MNIST training images, and
      prepares an expanded training set, with $250,000$ training images. We
      can then use those training images to train our network. We'll use
      the same network as above, with rectified linear units. In my initial
      experiments I reduced the number of training epochs - this made
      sense, since we're training with $5$ times as much data. But, in
      fact, expanding the data turned out to considerably reduce the effect
      of overfitting. And so, after some experimentation, I eventually went
      back to training for $60$ epochs. In any case, let's train:</p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">expanded_training_data</span><span class="p">,</span> <span class="n">_</span><span class="p">,</span> <span class="n">_</span> <span class="o">=</span> <span class="n">network3</span><span class="o">.</span><span class="n">load_data_shared</span><span class="p">(</span>
        <span class="s2">&quot;../data/mnist_expanded.pkl.gz&quot;</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">40</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="mi">4</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">expanded_training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">,</span> <span class="n">lmbda</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p></p>
    <p>Using the expanded training data I obtained a $99.37$ percent training
      accuracy. So this almost trivial change gives a substantial
      improvement in classification accuracy. Indeed, as we
      <a href="chap3.html#other_techniques_for_regularization">discussed
        earlier</a> this idea of algorithmically expanding the data can be
      taken further. Just to remind you of the flavour of some of the
      results in that earlier discussion: in 2003 Simard, Steinkraus and
      Platt*<span class="marginnote">
        *<a href="http://dx.doi.org/10.1109/ICDAR.2003.1227801">Best
          Practices for Convolutional Neural Networks Applied to Visual
          Document Analysis</a>, by Patrice Simard, Dave Steinkraus, and John
        Platt (2003).</span> improved their MNIST performance to $99.6$ percent
      using a neural network otherwise very similar to ours, using two
      convolutional-pooling layers, followed by a hidden fully-connected
      layer with $100$ neurons. There were a few differences of detail in
      their architecture - they didn't have the advantage of using
      rectified linear units, for instance - but the key to their improved
      performance was expanding the training data. They did this by
      rotating, translating, and skewing the MNIST training images. They
      also developed a process of "elastic distortion", a way of emulating
      the random oscillations hand muscles undergo when a person is writing.
      By combining all these processes they substantially increased the
      effective size of their training data, and that's how they achieved
      $99.6$ percent accuracy.
    </p>
    <p>
    <h4><a name="problem_437600"></a><a href="#problem_437600">Problem</a></h4>
    <ul>
      <li> The idea of convolutional layers is to behave in an invariant
        way across images. It may seem surprising, then, that our network
        can learn more when all we've done is translate the input data. Can
        you explain why this is actually quite reasonable?
    </ul>
    </p>
    <p><strong>Inserting an extra fully-connected layer:</strong> Can we do even
      better? One possibility is to use exactly the same procedure as
      above, but to expand the size of the fully-connected layer. I tried
      with $300$ and $1,000$ neurons, obtaining results of $99.46$ and
      $99.43$ percent, respectively. That's interesting, but not really a
      convincing win over the earlier result ($99.37$ percent).</p>
    <p>What about adding an extra fully-connected layer? Let's try inserting
      an extra fully-connected layer, so that we have two $100$-hidden
      neuron fully-connected layers:
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">40</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="mi">4</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">100</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">)],</span> <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">expanded_training_data</span><span class="p">,</span> <span class="mi">60</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">,</span> <span class="n">lmbda</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p>Doing this, I obtained a test accuracy of $99.43$ percent. Again, the
      expanded net isn't helping so much. Running similar experiments with
      fully-connected layers containing $300$ and $1,000$ neurons yields
      results of $99.48$ and $99.47$ percent. That's encouraging, but still
      falls short of a really decisive win.</p>
    <p><a name="final_conv"></a></p>
    <p>What's going on here? Is it that the expanded or extra
      fully-connected layers really don't help with MNIST? Or might it be
      that our network has the capacity to do better, but we're going about
      learning the wrong way? For instance, maybe we could use stronger
      regularization techniques to reduce the tendency to overfit. One
      possibility is the
      <a href="chap3.html#other_techniques_for_regularization">dropout</a>
      technique introduced back in Chapter 3. Recall that the basic idea of
      dropout is to remove individual activations at random while training
      the network. This makes the model more robust to the loss of
      individual pieces of evidence, and thus less likely to rely on
      particular idiosyncracies of the training data. Let's try applying
      dropout to the final fully-connected layers:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="o">&gt;&gt;&gt;</span> <span class="n">net</span> <span class="o">=</span> <span class="n">Network</span><span class="p">([</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">28</span><span class="p">,</span> <span class="mi">28</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">20</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">ConvPoolLayer</span><span class="p">(</span><span class="n">image_shape</span><span class="o">=</span><span class="p">(</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">12</span><span class="p">,</span> <span class="mi">12</span><span class="p">),</span> 
                      <span class="n">filter_shape</span><span class="o">=</span><span class="p">(</span><span class="mi">40</span><span class="p">,</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">),</span> 
                      <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span> 
                      <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span>
            <span class="n">n_in</span><span class="o">=</span><span class="mi">40</span><span class="o">*</span><span class="mi">4</span><span class="o">*</span><span class="mi">4</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.5</span><span class="p">),</span>
        <span class="n">FullyConnectedLayer</span><span class="p">(</span>
            <span class="n">n_in</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">ReLU</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.5</span><span class="p">),</span>
        <span class="n">SoftmaxLayer</span><span class="p">(</span><span class="n">n_in</span><span class="o">=</span><span class="mi">1000</span><span class="p">,</span> <span class="n">n_out</span><span class="o">=</span><span class="mi">10</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.5</span><span class="p">)],</span> 
        <span class="n">mini_batch_size</span><span class="p">)</span>
<span class="o">&gt;&gt;&gt;</span> <span class="n">net</span><span class="o">.</span><span class="n">SGD</span><span class="p">(</span><span class="n">expanded_training_data</span><span class="p">,</span> <span class="mi">40</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="mf">0.03</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p>Using this, we obtain an accuracy of $99.60$ percent, which is a
      substantial improvement over our earlier results, especially our main
      benchmark, the network with $100$ hidden neurons, where we achieved
      $99.37$ percent.</p>
    <p>There are two changes worth noting. </p>
    <p>First, I reduced the number of training epochs to $40$: dropout
      reduced overfitting, and so we learned faster.</p>
    <p>Second, the fully-connected hidden layers have $1,000$ neurons, not
      the $100$ used earlier. Of course, dropout effectively omits many of
      the neurons while training, so some expansion is to be expected. In
      fact, I tried experiments with both $300$ and $1,000$ hidden neurons,
      and obtained (very slightly) better validation performance with
      $1,000$ hidden neurons.</p>
    <p><strong>Using an ensemble of networks:</strong> An easy way to improve
      performance still further is to create several neural networks, and
      then get them to vote to determine the best classification. Suppose,
      for example, that we trained $5$ different neural networks using the
      prescription above, with each achieving accuracies near to $99.6$
      percent. Even though the networks would all have similar accuracies,
      they might well make different errors, due to the different random
      initializations. It's plausible that taking a vote amongst our $5$
      networks might yield a classification better than any individual
      network.</p>
    <p>This sounds too good to be true, but this kind of ensembling is a
      common trick with both neural networks and other machine learning
      techniques. And it does in fact yield further improvements: we end up
      with $99.67$ percent accuracy. In other words, our ensemble of
      networks classifies all but $33$ of the $10,000$ test images
      correctly. </p>
    <p>The remaining errors in the test set are shown below. The label in
      the top right is the correct classification, according to the MNIST
      data, while in the bottom right is the label output by our ensemble of
      nets:</p>
    <p>
      <center><img src="images/ensemble_errors.png" width="580px"></center>
    </p>
    <p>It's worth looking through these in detail. The first two digits, a 6
      and a 5, are genuine errors by our ensemble. However, they're also
      understandable errors, the kind a human could plausibly make. That 6
      really does look a lot like a 0, and the 5 looks a lot like a 3. The
      third image, supposedly an 8, actually looks to me more like a 9. So
      I'm siding with the network ensemble here: I think it's done a better
      job than whoever originally drew the digit. On the other hand, the
      fourth image, the 6, really does seem to be classified badly by our
      networks.</p>
    <p>And so on. In most cases our networks' choices seem at least
      plausible, and in some cases they've done a better job classifying
      than the original person did writing the digit. Overall, our networks
      offer exceptional performance, especially when you consider that they
      correctly classified 9,967 images which aren't shown. In that
      context, the few clear errors here seem quite understandable. Even a
      careful human makes the occasional mistake. And so I expect that only
      an extremely careful and methodical human would do much better. Our
      network is getting near to human performance.</p>
    <p><strong>Why we only applied dropout to the fully-connected layers:</strong> If
      you look carefully at the code above, you'll notice that we applied
      dropout only to the fully-connected section of the network, not to the
      convolutional layers. In principle we could apply a similar procedure
      to the convolutional layers. But, in fact, there's no need: the
      convolutional layers have considerable inbuilt resistance to
      overfitting. The reason is that the shared weights mean that
      convolutional filters are forced to learn from across the entire
      image. This makes them less likely to pick up on local idiosyncracies
      in the training data. And so there is less need to apply other
      regularizers, such as dropout.</p>
    <p><strong>Going further:</strong> It's possible to improve performance on MNIST
      still further. Rodrigo Benenson has compiled an
      <a href="http://rodrigob.github.io/are_we_there_yet/build/classification_datasets_results.html">informative
        summary page</a>, showing progress over the years, with links to
      papers. Many of these papers use deep convolutional networks along
      lines similar to the networks we've been using. If you dig through
      the papers you'll find many interesting techniques, and you may enjoy
      implementing some of them. If you do so it's wise to start
      implementation with a simple network that can be trained quickly,
      which will help you more rapidly understand what is going on.
    </p>
    <p>For the most part, I won't try to survey this recent work. But I
      can't resist making one exception. It's a 2010 paper by Cireșan,
      Meier, Gambardella, and
      Schmidhuber*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1003.0358">Deep, Big,
          Simple Neural Nets Excel on Handwritten Digit Recognition</a>, by Dan
        Claudiu Cireșan, Ueli Meier, Luca Maria Gambardella, and Jürgen
        Schmidhuber (2010).</span>. What I like about this paper is how simple it
      is. The network is a many-layer neural network, using only
      fully-connected layers (no convolutions). Their most successful
      network had hidden layers containing $2,500$, $2,000$, $1,500$,
      $1,000$, and $500$ neurons, respectively. They used ideas similar to
      Simard <em>et al</em> to expand their training data. But apart from
      that, they used few other tricks, including no convolutional layers:
      it was a plain, vanilla network, of the kind that, with enough
      patience, could have been trained in the 1980s (if the MNIST data set
      had existed), given enough computing power. They achieved a
      classification accuracy of $99.65$ percent, more or less the same as
      ours. The key was to use a very large, very deep network, and to use
      a GPU to speed up training. This let them train for many epochs.
      They also took advantage of their long training times to gradually
      decrease the learning rate from $10^{-3}$ to $10^{-6}$. It's a fun
      exercise to try to match these results using an architecture like
      theirs.</p>
    <p><strong>Why are we able to train?</strong> We saw in <a href="chap5.html">the
        last chapter</a> that there are fundamental obstructions to training in
      deep, many-layer neural networks. In particular, we saw that the
      gradient tends to be quite unstable: as we move from the output layer
      to earlier layers the gradient tends to either vanish (the vanishing
      gradient problem) or explode (the exploding gradient problem). Since
      the gradient is the signal we use to train, this causes problems.</p>
    <p>How have we avoided those results? </p>
    <p>Of course, the answer is that we haven't avoided these results.
      Instead, we've done a few things that help us proceed anyway. In
      particular: (1) Using convolutional layers greatly reduces the number
      of parameters in those layers, making the learning problem much
      easier; (2) Using more powerful regularization techniques (notably
      dropout and convolutional layers) to reduce overfitting, which is
      otherwise more of a problem in more complex networks; (3) Using
      rectified linear units instead of sigmoid neurons, to speed up
      training - empirically, often by a factor of $3$-$5$; (4) Using GPUs
      and being willing to train for a long period of time. In particular,
      in our final experiments we trained for $40$ epochs using a data set
      $5$ times larger than the raw MNIST training data. Earlier in the
      book we mostly trained for $30$ epochs using just the raw training
      data. Combining factors (3) and (4) it's as though we've trained a
      factor perhaps $30$ times longer than before.</p>
    <p>Your response may be "Is that it? Is that all we had to do to train
      deep networks? What's all the fuss about?"</p>
    <p>Of course, we've used other ideas, too: making use of sufficiently
      large data sets (to help avoid overfitting); using the right cost
      function (to
      <a href="chap3.html#the_cross\-entropy_cost_function">avoid a
        learning slowdown</a>); using
      <a href="chap3.html#weight_initialization">good weight initializations</a>
      (also to avoid a learning slowdown, due to neuron saturation);
      <a href="chap3.html#other_techniques_for_regularization">algorithmically
        expanding the training data</a>. We discussed these and other ideas in
      earlier chapters, and have for the most part been able to reuse these
      ideas with little comment in this chapter.
    </p>
    <p>With that said, this really is a rather simple set of ideas. Simple,
      but powerful, when used in concert. Getting started with deep
      learning has turned out to be pretty easy!</p>
    <p><strong>How deep are these networks, anyway?</strong> Counting the
      convolutional-pooling layers as single layers, our final architecture
      has $4$ hidden layers. Does such a network really deserve to be
      called a <em>deep</em> network? Of course, $4$ hidden layers is many
      more than in the shallow networks we studied earlier. Most of those
      networks only had a single hidden layer, or occasionally $2$ hidden
      layers. On the other hand, as of 2015 state-of-the-art deep networks
      sometimes have dozens of hidden layers. I've occasionally heard
      people adopt a deeper-than-thou attitude, holding that if you're not
      keeping-up-with-the-Joneses in terms of number of hidden layers, then
      you're not really doing deep learning. I'm not sympathetic to this
      attitude, in part because it makes the definition of deep learning
      into something which depends upon the result-of-the-moment. The real
      breakthrough in deep learning was to realize that it's practical to go
      beyond the shallow $1$- and $2$-hidden layer networks that dominated
      work until the mid-2000s. That really was a significant breakthrough,
      opening up the exploration of much more expressive models. But beyond
      that, the number of layers is not of primary fundamental interest.
      Rather, the use of deeper networks is a tool to use to help achieve
      other goals - like better classification accuracies.</p>
    <p><strong>A word on procedure:</strong> In this section, we've smoothly moved
      from single hidden-layer shallow networks to many-layer convolutional
      networks. It all seemed so easy! We make a change and, for the
      most part, we get an improvement. If you start experimenting, I can
      guarantee things won't always be so smooth. The reason is that I've
      presented a cleaned-up narrative, omitting many experiments -
      including many failed experiments. This cleaned-up narrative will
      hopefully help you get clear on the basic ideas. But it also runs the
      risk of conveying an incomplete impression. Getting a good, working
      network can involve a lot of trial and error, and occasional
      frustration. In practice, you should expect to engage in quite a bit
      of experimentation. To speed that process up you may find it helpful
      to revisit Chapter 3's discussion of
      <a href="chap3.html#how_to_choose_a_neural_network's_hyper-parameters">how
        to choose a neural network's hyper-parameters</a>, and perhaps also to
      look at some of the further reading suggested in that section.
    </p>
    <p></p>
    <p></p>
    <p></p>
    <p>
    <h3><a name="the_code_for_our_convolutional_networks"></a><a href="#the_code_for_our_convolutional_networks">The
        code for our convolutional networks</a></h3>
    </p>
    <p>Alright, let's take a look at the code for our program,
      <tt>network3.py</tt>. Structurally, it's similar to <tt>network2.py</tt>,
      the program we developed in <a href="chap3.html">Chapter 3</a>, although the
      details differ, due to the use of Theano. We'll start by looking at
      the <tt>FullyConnectedLayer</tt> class, which is similar to the layers
      studied earlier in the book. Here's the code (discussion
      below)*<span class="marginnote">
        *Note added November 2016: several readers have noted
        that in the line initializing <tt>self.w</tt>, I set
        <tt>scale=np.sqrt(1.0/n_out)</tt>, when the arguments of Chapter 3
        suggest a better initialization may be
        <tt>scale=np.sqrt(1.0/n_in)</tt>. This was simply a mistake on my
        part. In an ideal world I'd rerun all the examples in this chapter
        with the correct code. Still, I've moved on to other projects, so am
        going to let the error go.</span>:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="k">class</span> <span class="nc">FullyConnectedLayer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>

    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span> <span class="o">=</span> <span class="n">n_in</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_out</span> <span class="o">=</span> <span class="n">n_out</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span> <span class="o">=</span> <span class="n">activation_fn</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span> <span class="o">=</span> <span class="n">p_dropout</span>
        <span class="c1"># Initialize weights and biases</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">w</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span>
                <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
                    <span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n_out</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">)),</span>
                <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_out</span><span class="p">,)),</span>
                       <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">]</span>

    <span class="k">def</span> <span class="nf">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inpt</span><span class="p">,</span> <span class="n">inpt_dropout</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt</span> <span class="o">=</span> <span class="n">inpt</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">))</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="p">(</span>
            <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span><span class="o">*</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span> <span class="o">=</span> <span class="n">dropout_layer</span><span class="p">(</span>
            <span class="n">inpt_dropout</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="p">(</span>
            <span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>

    <span class="k">def</span> <span class="nf">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
        <span class="s2">&quot;Return the accuracy for the mini-batch.&quot;</span>
        <span class="k">return</span> <span class="n">T</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span><span class="p">))</span>
</pre>
    </div>
    </p>
    <p>Much of the <tt>__init__</tt> method is self-explanatory, but a few
      remarks may help clarify the code. As per usual, we randomly
      initialize the weights and biases as normal random variables with
      suitable standard deviations. The lines doing this look a little
      forbidding. However, most of the complication is just loading the
      weights and biases into what Theano calls shared variables. This
      ensures that these variables can be processed on the GPU, if one is
      available. We won't get too much into the details of this. If you're
      interested, you can dig into the
      <a href="http://deeplearning.net/software/theano/index.html">Theano
        documentation</a>. Note also that this weight and bias initialization
      is designed for the sigmoid activation function (as
      <a href="chap3.html#weight_initialization">discussed earlier</a>).
      Ideally, we'd initialize the weights and biases somewhat differently
      for activation functions such as the tanh and rectified linear
      function. This is discussed further in problems below. The
      <tt>__init__</tt> method finishes with
      <tt>self.params = [self.w, self.b]</tt>. This is a handy way to bundle
      up all the learnable parameters associated to the layer. Later on,
      the <tt>Network.SGD</tt> method will use <tt>params</tt> attributes to
      figure out what variables in a <tt>Network</tt> instance can learn.
    </p>
    <p>The <tt>set_inpt</tt> method is used to set the input to the layer, and
      to compute the corresponding output. I use the name <tt>inpt</tt>
      rather than <tt>input</tt> because <tt>input</tt> is a built-in function
      in Python, and messing with built-ins tends to cause unpredictable
      behavior and difficult-to-diagnose bugs. Note that we actually set
      the input in two separate ways: as <tt>self.inpt</tt> and
      <tt>self.inpt_dropout</tt>. This is done because during training we may
      want to use dropout. If that's the case then we want to remove a
      fraction <tt>self.p_dropout</tt> of the neurons. That's what the
      function <tt>dropout_layer</tt> in the second-last line of the
      <tt>set_inpt</tt> method is doing. So <tt>self.inpt_dropout</tt> and
      <tt>self.output_dropout</tt> are used during training, while
      <tt>self.inpt</tt> and <tt>self.output</tt> are used for all other
      purposes, e.g., evaluating accuracy on the validation and test data.
    </p>
    <p>The <tt>ConvPoolLayer</tt> and <tt>SoftmaxLayer</tt> class definitions are
      similar to <tt>FullyConnectedLayer</tt>. Indeed, they're so close that
      I won't excerpt the code here. If you're interested you can look at
      the full listing for <tt>network3.py</tt>, later in this section.</p>
    <p>However, a couple of minor differences of detail are worth mentioning.
      Most obviously, in both <tt>ConvPoolLayer</tt> and <tt>SoftmaxLayer</tt>
      we compute the output activations in the way appropriate to that layer
      type. Fortunately, Theano makes that easy, providing built-in
      operations to compute convolutions, max-pooling, and the softmax
      function.</p>
    <p>Less obviously, when we <a href="chap3.html#softmax">introduced the
        softmax layer</a>, we never discussed how to initialize the weights and
      biases. Elsewhere we've argued that for sigmoid layers we should
      initialize the weights using suitably parameterized normal random
      variables. But that heuristic argument was specific to sigmoid
      neurons (and, with some amendment, to tanh neurons). However, there's
      no particular reason the argument should apply to softmax layers. So
      there's no <em>a priori</em> reason to apply that initialization again.
      Rather than do that, I shall initialize all the weights and biases to
      be $0$. This is a rather <em>ad hoc</em> procedure, but works well
      enough in practice.</p>
    <p>Okay, we've looked at all the layer classes. What about the
      <tt>Network</tt> class? Let's start by looking at the <tt>__init__</tt>
      method:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="k">class</span> <span class="nc">Network</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
    
    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">layers</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Takes a list of `layers`, describing the network architecture, and</span>
<span class="sd">        a value for the `mini_batch_size` to be used during training</span>
<span class="sd">        by stochastic gradient descent.</span>

<span class="sd">        &quot;&quot;&quot;</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">layers</span> <span class="o">=</span> <span class="n">layers</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span> <span class="o">=</span> <span class="n">mini_batch_size</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="n">param</span> <span class="k">for</span> <span class="n">layer</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span> <span class="k">for</span> <span class="n">param</span> <span class="ow">in</span> <span class="n">layer</span><span class="o">.</span><span class="n">params</span><span class="p">]</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>  
        <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">ivector</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
        <span class="n">init_layer</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
        <span class="n">init_layer</span><span class="o">.</span><span class="n">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">)</span>
        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">)):</span>
            <span class="n">prev_layer</span><span class="p">,</span> <span class="n">layer</span>  <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
            <span class="n">layer</span><span class="o">.</span><span class="n">set_inpt</span><span class="p">(</span>
                <span class="n">prev_layer</span><span class="o">.</span><span class="n">output</span><span class="p">,</span> <span class="n">prev_layer</span><span class="o">.</span><span class="n">output_dropout</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">output</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">output_dropout</span>
</pre>
    </div>
    </p>
    <p>Most of this is self-explanatory, or nearly so. The line
      <tt>self.params = [param for layer in ...]</tt> bundles up the
      parameters for each layer into a single list. As anticipated above,
      the <tt>Network.SGD</tt> method will use <tt>self.params</tt> to figure
      out what variables in the <tt>Network</tt> can learn. The lines
      <tt>self.x = T.matrix("x")</tt> and <tt>self.y = T.ivector("y")</tt>
      define Theano symbolic variables named <tt>x</tt> and <tt>y</tt>. These
      will be used to represent the input and desired output from the
      network.
    </p>
    <p>Now, this isn't a Theano tutorial, and so we won't get too deeply into
      what it means that these are symbolic variables*<span class="marginnote">
        *The
        <a href="http://deeplearning.net/software/theano/index.html">Theano
          documentation</a> provides a good introduction to Theano. And if you
        get stuck, you may find it helpful to look at one of the other
        tutorials available online. For instance,
        <a href="http://nbviewer.ipython.org/github/craffel/theano-tutorial/blob/master/Theano%20Tutorial.ipynb">this
          tutorial</a> covers many basics.</span>. But the rough idea is that these
      represent mathematical variables, <em>not</em> explicit values. We can
      do all the usual things one would do with such variables: add,
      subtract, and multiply them, apply functions, and so on. Indeed,
      Theano provides many ways of manipulating such symbolic variables,
      doing things like convolutions, max-pooling, and so on. But the big
      win is the ability to do fast symbolic differentiation, using a very
      general form of the backpropagation algorithm. This is extremely
      useful for applying stochastic gradient descent to a wide variety of
      network architectures. In particular, the next few lines of code
      define symbolic outputs from the network. We start by setting the
      input to the initial layer, with the line</p>
    <p>
    <div class="highlight">
      <pre><span></span>        <span class="n">init_layer</span><span class="o">.</span><span class="n">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p>Note that the inputs are set one mini-batch at a time, which is why
      the mini-batch size is there. Note also that we pass the input
      <tt>self.x</tt> in twice: this is because we may use the network in two
      different ways (with or without dropout). The <tt>for</tt> loop then
      propagates the symbolic variable <tt>self.x</tt> forward through the
      layers of the <tt>Network</tt>. This allows us to define the final
      <tt>output</tt> and <tt>output_dropout</tt> attributes, which symbolically
      represent the output from the <tt>Network</tt>.
    </p>
    <p>Now that we've understood how a <tt>Network</tt> is initialized, let's
      look at how it is trained, using the <tt>SGD</tt> method. The code
      looks lengthy, but its structure is actually rather simple.
      Explanatory comments after the code.</p>
    <p>
    <div class="highlight">
      <pre><span></span>    <span class="k">def</span> <span class="nf">SGD</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">training_data</span><span class="p">,</span> <span class="n">epochs</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span> 
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">,</span> <span class="n">lmbda</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Train the network using mini-batch stochastic gradient descent.&quot;&quot;&quot;</span>
        <span class="n">training_x</span><span class="p">,</span> <span class="n">training_y</span> <span class="o">=</span> <span class="n">training_data</span>
        <span class="n">validation_x</span><span class="p">,</span> <span class="n">validation_y</span> <span class="o">=</span> <span class="n">validation_data</span>
        <span class="n">test_x</span><span class="p">,</span> <span class="n">test_y</span> <span class="o">=</span> <span class="n">test_data</span>

        <span class="c1"># compute number of minibatches for training, validation and testing</span>
        <span class="n">num_training_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">training_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>
        <span class="n">num_validation_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">validation_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>
        <span class="n">num_test_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">test_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>

        <span class="c1"># define the (regularized) cost function, symbolic gradients, and updates</span>
        <span class="n">l2_norm_squared</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">layer</span><span class="o">.</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="k">for</span> <span class="n">layer</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">])</span>
        <span class="n">cost</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cost</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">+</span>\
               <span class="mf">0.5</span><span class="o">*</span><span class="n">lmbda</span><span class="o">*</span><span class="n">l2_norm_squared</span><span class="o">/</span><span class="n">num_training_batches</span>
        <span class="n">grads</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">cost</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
        <span class="n">updates</span> <span class="o">=</span> <span class="p">[(</span><span class="n">param</span><span class="p">,</span> <span class="n">param</span><span class="o">-</span><span class="n">eta</span><span class="o">*</span><span class="n">grad</span><span class="p">)</span> 
                   <span class="k">for</span> <span class="n">param</span><span class="p">,</span> <span class="n">grad</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">,</span> <span class="n">grads</span><span class="p">)]</span>

        <span class="c1"># define functions to train a mini-batch, and to compute the</span>
        <span class="c1"># accuracy in validation and test mini-batches.</span>
        <span class="n">i</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">lscalar</span><span class="p">()</span> <span class="c1"># mini-batch index</span>
        <span class="n">train_mb</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">cost</span><span class="p">,</span> <span class="n">updates</span><span class="o">=</span><span class="n">updates</span><span class="p">,</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
                <span class="n">training_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span> 
                <span class="n">training_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="n">validate_mb_accuracy</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">),</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span> 
                <span class="n">validation_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span> 
                <span class="n">validation_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="n">test_mb_accuracy</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">),</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span> 
                <span class="n">test_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span> 
                <span class="n">test_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">test_mb_predictions</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y_out</span><span class="p">,</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span> 
                <span class="n">test_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="c1"># Do the actual training</span>
        <span class="n">best_validation_accuracy</span> <span class="o">=</span> <span class="mf">0.0</span>
        <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
            <span class="k">for</span> <span class="n">minibatch_index</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_training_batches</span><span class="p">):</span>
                <span class="n">iteration</span> <span class="o">=</span> <span class="n">num_training_batches</span><span class="o">*</span><span class="n">epoch</span><span class="o">+</span><span class="n">minibatch_index</span>
                <span class="k">if</span> <span class="n">iteration</span> 
                    <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Training mini-batch number {0}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">iteration</span><span class="p">))</span>
                <span class="n">cost_ij</span> <span class="o">=</span> <span class="n">train_mb</span><span class="p">(</span><span class="n">minibatch_index</span><span class="p">)</span>
                <span class="k">if</span> <span class="p">(</span><span class="n">iteration</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> 
                    <span class="n">validation_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span>
                        <span class="p">[</span><span class="n">validate_mb_accuracy</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_validation_batches</span><span class="p">)])</span>
                    <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Epoch {0}: validation accuracy {1:.2</span>
                        <span class="n">epoch</span><span class="p">,</span> <span class="n">validation_accuracy</span><span class="p">))</span>
                    <span class="k">if</span> <span class="n">validation_accuracy</span> <span class="o">&gt;=</span> <span class="n">best_validation_accuracy</span><span class="p">:</span>
                        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;This is the best validation accuracy to date.&quot;</span><span class="p">)</span>
                        <span class="n">best_validation_accuracy</span> <span class="o">=</span> <span class="n">validation_accuracy</span>
                        <span class="n">best_iteration</span> <span class="o">=</span> <span class="n">iteration</span>
                        <span class="k">if</span> <span class="n">test_data</span><span class="p">:</span>
                            <span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span>
                                <span class="p">[</span><span class="n">test_mb_accuracy</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_test_batches</span><span class="p">)])</span>
                            <span class="k">print</span><span class="p">(</span><span class="s1">&#39;The corresponding test accuracy is {0:.2</span>
                                <span class="n">test_accuracy</span><span class="p">))</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Finished training network.&quot;</span><span class="p">)</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Best validation accuracy of {0:.2</span>
            <span class="n">best_validation_accuracy</span><span class="p">,</span> <span class="n">best_iteration</span><span class="p">))</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Corresponding test accuracy of {0:.2</span>
</pre>
    </div>
    </p>
    <p>The first few lines are straightforward, separating the datasets into
      $x$ and $y$ components, and computing the number of mini-batches used
      in each dataset. The next few lines are more interesting, and show
      some of what makes Theano fun to work with. Let's explicitly excerpt
      the lines here:</p>
    <p>
    <div class="highlight">
      <pre><span></span>        <span class="c1"># define the (regularized) cost function, symbolic gradients, and updates</span>
        <span class="n">l2_norm_squared</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">layer</span><span class="o">.</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="k">for</span> <span class="n">layer</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">])</span>
        <span class="n">cost</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cost</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">+</span>\
               <span class="mf">0.5</span><span class="o">*</span><span class="n">lmbda</span><span class="o">*</span><span class="n">l2_norm_squared</span><span class="o">/</span><span class="n">num_training_batches</span>
        <span class="n">grads</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">cost</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
        <span class="n">updates</span> <span class="o">=</span> <span class="p">[(</span><span class="n">param</span><span class="p">,</span> <span class="n">param</span><span class="o">-</span><span class="n">eta</span><span class="o">*</span><span class="n">grad</span><span class="p">)</span> 
                   <span class="k">for</span> <span class="n">param</span><span class="p">,</span> <span class="n">grad</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">,</span> <span class="n">grads</span><span class="p">)]</span>
</pre>
    </div>
    </p>
    <p>In these lines we symbolically set up the regularized log-likelihood
      cost function, compute the corresponding derivatives in the gradient
      function, as well as the corresponding parameter updates. Theano lets
      us achieve all of this in just these few lines. The only thing hidden
      is that computing the <tt>cost</tt> involves a call to the <tt>cost</tt>
      method for the output layer; that code is elsewhere in
      <tt>network3.py</tt>. But that code is short and simple, anyway. With
      all these things defined, the stage is set to define the
      <tt>train_mb</tt> function, a Theano symbolic function which uses the
      <tt>updates</tt> to update the <tt>Network</tt> parameters, given a
      mini-batch index. Similarly, <tt>validate_mb_accuracy</tt> and
      <tt>test_mb_accuracy</tt> compute the accuracy of the <tt>Network</tt> on
      any given mini-batch of validation or test data. By averaging over
      these functions, we will be able to compute accuracies on the entire
      validation and test data sets.
    </p>
    <p>The remainder of the <tt>SGD</tt> method is self-explanatory - we
      simply iterate over the epochs, repeatedly training the network on
      mini-batches of training data, and computing the validation and test
      accuracies. </p>
    <p>Okay, we've now understood the most important pieces of code in
      <tt>network3.py</tt>. Let's take a brief look at the entire program.
      You don't need to read through this in detail, but you may enjoy
      glancing over it, and perhaps diving down into any pieces that strike
      your fancy. The best way to really understand it is, of course, by
      modifying it, adding extra features, or refactoring anything you think
      could be done more elegantly. After the code, there are some problems
      which contain a few starter suggestions for things to do. Here's the
      code*<span class="marginnote">
        *Using Theano on a GPU can be a little tricky. In
        particular, it's easy to make the mistake of pulling data off the
        GPU, which can slow things down a lot. I've tried to avoid this.
        With that said, this code can certainly be sped up quite a bit
        further with careful optimization of Theano's configuration. See
        the Theano documentation for more details.</span>:
    </p>
    <p>
    <div class="highlight">
      <pre><span></span><span class="sd">&quot;&quot;&quot;network3.py</span>
<span class="sd">~~~~~~~~~~~~~~</span>

<span class="sd">A Theano-based program for training and running simple neural</span>
<span class="sd">networks.</span>

<span class="sd">Supports several layer types (fully connected, convolutional, max</span>
<span class="sd">pooling, softmax), and activation functions (sigmoid, tanh, and</span>
<span class="sd">rectified linear units, with more easily added).</span>

<span class="sd">When run on a CPU, this program is much faster than network.py and</span>
<span class="sd">network2.py.  However, unlike network.py and network2.py it can also</span>
<span class="sd">be run on a GPU, which makes it faster still.</span>

<span class="sd">Because the code is based on Theano, the code is different in many</span>
<span class="sd">ways from network.py and network2.py.  However, where possible I have</span>
<span class="sd">tried to maintain consistency with the earlier programs.  In</span>
<span class="sd">particular, the API is similar to network2.py.  Note that I have</span>
<span class="sd">focused on making the code simple, easily readable, and easily</span>
<span class="sd">modifiable.  It is not optimized, and omits many desirable features.</span>

<span class="sd">This program incorporates ideas from the Theano documentation on</span>
<span class="sd">convolutional neural nets (notably,</span>
<span class="sd">http://deeplearning.net/tutorial/lenet.html ), from Misha Denil&#39;s</span>
<span class="sd">implementation of dropout (https://github.com/mdenil/dropout ), and</span>
<span class="sd">from Chris Olah (http://colah.github.io ).</span>

<span class="sd">Written for Theano 0.6 and 0.7, needs some changes for more recent</span>
<span class="sd">versions of Theano.</span>

<span class="sd">&quot;&quot;&quot;</span>

<span class="c1">#### Libraries</span>
<span class="c1"># Standard library</span>
<span class="kn">import</span> <span class="nn">cPickle</span>
<span class="kn">import</span> <span class="nn">gzip</span>

<span class="c1"># Third-party libraries</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">theano</span>
<span class="kn">import</span> <span class="nn">theano.tensor</span> <span class="kn">as</span> <span class="nn">T</span>
<span class="kn">from</span> <span class="nn">theano.tensor.nnet</span> <span class="kn">import</span> <span class="n">conv</span>
<span class="kn">from</span> <span class="nn">theano.tensor.nnet</span> <span class="kn">import</span> <span class="n">softmax</span>
<span class="kn">from</span> <span class="nn">theano.tensor</span> <span class="kn">import</span> <span class="n">shared_randomstreams</span>
<span class="kn">from</span> <span class="nn">theano.tensor.signal</span> <span class="kn">import</span> <span class="n">downsample</span>

<span class="c1"># Activation functions for neurons</span>
<span class="k">def</span> <span class="nf">linear</span><span class="p">(</span><span class="n">z</span><span class="p">):</span> <span class="k">return</span> <span class="n">z</span>
<span class="k">def</span> <span class="nf">ReLU</span><span class="p">(</span><span class="n">z</span><span class="p">):</span> <span class="k">return</span> <span class="n">T</span><span class="o">.</span><span class="n">maximum</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">z</span><span class="p">)</span>
<span class="kn">from</span> <span class="nn">theano.tensor.nnet</span> <span class="kn">import</span> <span class="n">sigmoid</span>
<span class="kn">from</span> <span class="nn">theano.tensor</span> <span class="kn">import</span> <span class="n">tanh</span>


<span class="c1">#### Constants</span>
<span class="n">GPU</span> <span class="o">=</span> <span class="bp">True</span>
<span class="k">if</span> <span class="n">GPU</span><span class="p">:</span>
    <span class="k">print</span> <span class="s2">&quot;Trying to run under a GPU.  If this is not desired, then modify &quot;</span><span class="o">+</span>\
        <span class="s2">&quot;network3.py</span><span class="se">\n</span><span class="s2">to set the GPU flag to False.&quot;</span>
    <span class="k">try</span><span class="p">:</span> <span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">device</span> <span class="o">=</span> <span class="s1">&#39;gpu&#39;</span>
    <span class="k">except</span><span class="p">:</span> <span class="k">pass</span> <span class="c1"># it&#39;s already set</span>
    <span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span> <span class="o">=</span> <span class="s1">&#39;float32&#39;</span>
<span class="k">else</span><span class="p">:</span>
    <span class="k">print</span> <span class="s2">&quot;Running with a CPU.  If this is not desired, then the modify &quot;</span><span class="o">+</span>\
        <span class="s2">&quot;network3.py to set</span><span class="se">\n</span><span class="s2">the GPU flag to True.&quot;</span>

<span class="c1">#### Load the MNIST data</span>
<span class="k">def</span> <span class="nf">load_data_shared</span><span class="p">(</span><span class="n">filename</span><span class="o">=</span><span class="s2">&quot;../data/mnist.pkl.gz&quot;</span><span class="p">):</span>
    <span class="n">f</span> <span class="o">=</span> <span class="n">gzip</span><span class="o">.</span><span class="n">open</span><span class="p">(</span><span class="n">filename</span><span class="p">,</span> <span class="s1">&#39;rb&#39;</span><span class="p">)</span>
    <span class="n">training_data</span><span class="p">,</span> <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span> <span class="o">=</span> <span class="n">cPickle</span><span class="o">.</span><span class="n">load</span><span class="p">(</span><span class="n">f</span><span class="p">)</span>
    <span class="n">f</span><span class="o">.</span><span class="n">close</span><span class="p">()</span>
    <span class="k">def</span> <span class="nf">shared</span><span class="p">(</span><span class="n">data</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Place the data into shared variables.  This allows Theano to copy</span>
<span class="sd">        the data to the GPU, if one is available.</span>

<span class="sd">        &quot;&quot;&quot;</span>
        <span class="n">shared_x</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="n">shared_y</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">data</span><span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="k">return</span> <span class="n">shared_x</span><span class="p">,</span> <span class="n">T</span><span class="o">.</span><span class="n">cast</span><span class="p">(</span><span class="n">shared_y</span><span class="p">,</span> <span class="s2">&quot;int32&quot;</span><span class="p">)</span>
    <span class="k">return</span> <span class="p">[</span><span class="n">shared</span><span class="p">(</span><span class="n">training_data</span><span class="p">),</span> <span class="n">shared</span><span class="p">(</span><span class="n">validation_data</span><span class="p">),</span> <span class="n">shared</span><span class="p">(</span><span class="n">test_data</span><span class="p">)]</span>

<span class="c1">#### Main class used to construct and train networks</span>
<span class="k">class</span> <span class="nc">Network</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>

    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">layers</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Takes a list of `layers`, describing the network architecture, and</span>
<span class="sd">        a value for the `mini_batch_size` to be used during training</span>
<span class="sd">        by stochastic gradient descent.</span>

<span class="sd">        &quot;&quot;&quot;</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">layers</span> <span class="o">=</span> <span class="n">layers</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span> <span class="o">=</span> <span class="n">mini_batch_size</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="n">param</span> <span class="k">for</span> <span class="n">layer</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span> <span class="k">for</span> <span class="n">param</span> <span class="ow">in</span> <span class="n">layer</span><span class="o">.</span><span class="n">params</span><span class="p">]</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">x</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">matrix</span><span class="p">(</span><span class="s2">&quot;x&quot;</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">y</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">ivector</span><span class="p">(</span><span class="s2">&quot;y&quot;</span><span class="p">)</span>
        <span class="n">init_layer</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>
        <span class="n">init_layer</span><span class="o">.</span><span class="n">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">)</span>
        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">)):</span>
            <span class="n">prev_layer</span><span class="p">,</span> <span class="n">layer</span>  <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="n">j</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="n">j</span><span class="p">]</span>
            <span class="n">layer</span><span class="o">.</span><span class="n">set_inpt</span><span class="p">(</span>
                <span class="n">prev_layer</span><span class="o">.</span><span class="n">output</span><span class="p">,</span> <span class="n">prev_layer</span><span class="o">.</span><span class="n">output_dropout</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">output</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">output_dropout</span>

    <span class="k">def</span> <span class="nf">SGD</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">training_data</span><span class="p">,</span> <span class="n">epochs</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">,</span> <span class="n">eta</span><span class="p">,</span>
            <span class="n">validation_data</span><span class="p">,</span> <span class="n">test_data</span><span class="p">,</span> <span class="n">lmbda</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;Train the network using mini-batch stochastic gradient descent.&quot;&quot;&quot;</span>
        <span class="n">training_x</span><span class="p">,</span> <span class="n">training_y</span> <span class="o">=</span> <span class="n">training_data</span>
        <span class="n">validation_x</span><span class="p">,</span> <span class="n">validation_y</span> <span class="o">=</span> <span class="n">validation_data</span>
        <span class="n">test_x</span><span class="p">,</span> <span class="n">test_y</span> <span class="o">=</span> <span class="n">test_data</span>

        <span class="c1"># compute number of minibatches for training, validation and testing</span>
        <span class="n">num_training_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">training_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>
        <span class="n">num_validation_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">validation_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>
        <span class="n">num_test_batches</span> <span class="o">=</span> <span class="n">size</span><span class="p">(</span><span class="n">test_data</span><span class="p">)</span><span class="o">/</span><span class="n">mini_batch_size</span>

        <span class="c1"># define the (regularized) cost function, symbolic gradients, and updates</span>
        <span class="n">l2_norm_squared</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">([(</span><span class="n">layer</span><span class="o">.</span><span class="n">w</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span> <span class="k">for</span> <span class="n">layer</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">])</span>
        <span class="n">cost</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">cost</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span><span class="o">+</span>\
               <span class="mf">0.5</span><span class="o">*</span><span class="n">lmbda</span><span class="o">*</span><span class="n">l2_norm_squared</span><span class="o">/</span><span class="n">num_training_batches</span>
        <span class="n">grads</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">grad</span><span class="p">(</span><span class="n">cost</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">)</span>
        <span class="n">updates</span> <span class="o">=</span> <span class="p">[(</span><span class="n">param</span><span class="p">,</span> <span class="n">param</span><span class="o">-</span><span class="n">eta</span><span class="o">*</span><span class="n">grad</span><span class="p">)</span>
                   <span class="k">for</span> <span class="n">param</span><span class="p">,</span> <span class="n">grad</span> <span class="ow">in</span> <span class="nb">zip</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">params</span><span class="p">,</span> <span class="n">grads</span><span class="p">)]</span>

        <span class="c1"># define functions to train a mini-batch, and to compute the</span>
        <span class="c1"># accuracy in validation and test mini-batches.</span>
        <span class="n">i</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">lscalar</span><span class="p">()</span> <span class="c1"># mini-batch index</span>
        <span class="n">train_mb</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">cost</span><span class="p">,</span> <span class="n">updates</span><span class="o">=</span><span class="n">updates</span><span class="p">,</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
                <span class="n">training_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
                <span class="n">training_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="n">validate_mb_accuracy</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">),</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
                <span class="n">validation_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
                <span class="n">validation_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="n">test_mb_accuracy</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">),</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
                <span class="n">test_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">],</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">y</span><span class="p">:</span>
                <span class="n">test_y</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">test_mb_predictions</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">function</span><span class="p">(</span>
            <span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="bp">self</span><span class="o">.</span><span class="n">layers</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">.</span><span class="n">y_out</span><span class="p">,</span>
            <span class="n">givens</span><span class="o">=</span><span class="p">{</span>
                <span class="bp">self</span><span class="o">.</span><span class="n">x</span><span class="p">:</span>
                <span class="n">test_x</span><span class="p">[</span><span class="n">i</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">:</span> <span class="p">(</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="bp">self</span><span class="o">.</span><span class="n">mini_batch_size</span><span class="p">]</span>
            <span class="p">})</span>
        <span class="c1"># Do the actual training</span>
        <span class="n">best_validation_accuracy</span> <span class="o">=</span> <span class="mf">0.0</span>
        <span class="k">for</span> <span class="n">epoch</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">epochs</span><span class="p">):</span>
            <span class="k">for</span> <span class="n">minibatch_index</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_training_batches</span><span class="p">):</span>
                <span class="n">iteration</span> <span class="o">=</span> <span class="n">num_training_batches</span><span class="o">*</span><span class="n">epoch</span><span class="o">+</span><span class="n">minibatch_index</span>
                <span class="k">if</span> <span class="n">iteration</span> <span class="o">%</span> <span class="mi">1000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
                    <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Training mini-batch number {0}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">iteration</span><span class="p">))</span>
                <span class="n">cost_ij</span> <span class="o">=</span> <span class="n">train_mb</span><span class="p">(</span><span class="n">minibatch_index</span><span class="p">)</span>
                <span class="k">if</span> <span class="p">(</span><span class="n">iteration</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="n">num_training_batches</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
                    <span class="n">validation_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span>
                        <span class="p">[</span><span class="n">validate_mb_accuracy</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_validation_batches</span><span class="p">)])</span>
                    <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Epoch {0}: validation accuracy {1:.2%}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
                        <span class="n">epoch</span><span class="p">,</span> <span class="n">validation_accuracy</span><span class="p">))</span>
                    <span class="k">if</span> <span class="n">validation_accuracy</span> <span class="o">&gt;=</span> <span class="n">best_validation_accuracy</span><span class="p">:</span>
                        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;This is the best validation accuracy to date.&quot;</span><span class="p">)</span>
                        <span class="n">best_validation_accuracy</span> <span class="o">=</span> <span class="n">validation_accuracy</span>
                        <span class="n">best_iteration</span> <span class="o">=</span> <span class="n">iteration</span>
                        <span class="k">if</span> <span class="n">test_data</span><span class="p">:</span>
                            <span class="n">test_accuracy</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span>
                                <span class="p">[</span><span class="n">test_mb_accuracy</span><span class="p">(</span><span class="n">j</span><span class="p">)</span> <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">xrange</span><span class="p">(</span><span class="n">num_test_batches</span><span class="p">)])</span>
                            <span class="k">print</span><span class="p">(</span><span class="s1">&#39;The corresponding test accuracy is {0:.2%}&#39;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
                                <span class="n">test_accuracy</span><span class="p">))</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Finished training network.&quot;</span><span class="p">)</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Best validation accuracy of {0:.2%} obtained at iteration {1}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span>
            <span class="n">best_validation_accuracy</span><span class="p">,</span> <span class="n">best_iteration</span><span class="p">))</span>
        <span class="k">print</span><span class="p">(</span><span class="s2">&quot;Corresponding test accuracy of {0:.2%}&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">test_accuracy</span><span class="p">))</span>

<span class="c1">#### Define layer types</span>

<span class="k">class</span> <span class="nc">ConvPoolLayer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>
    <span class="sd">&quot;&quot;&quot;Used to create a combination of a convolutional and a max-pooling</span>
<span class="sd">    layer.  A more sophisticated implementation would separate the</span>
<span class="sd">    two, but for our purposes we&#39;ll always use them together, and it</span>
<span class="sd">    simplifies the code, so it makes sense to combine them.</span>

<span class="sd">    &quot;&quot;&quot;</span>

    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">filter_shape</span><span class="p">,</span> <span class="n">image_shape</span><span class="p">,</span> <span class="n">poolsize</span><span class="o">=</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">),</span>
                 <span class="n">activation_fn</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">):</span>
        <span class="sd">&quot;&quot;&quot;`filter_shape` is a tuple of length 4, whose entries are the number</span>
<span class="sd">        of filters, the number of input feature maps, the filter height, and the</span>
<span class="sd">        filter width.</span>

<span class="sd">        `image_shape` is a tuple of length 4, whose entries are the</span>
<span class="sd">        mini-batch size, the number of input feature maps, the image</span>
<span class="sd">        height, and the image width.</span>

<span class="sd">        `poolsize` is a tuple of length 2, whose entries are the y and</span>
<span class="sd">        x pooling sizes.</span>

<span class="sd">        &quot;&quot;&quot;</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">filter_shape</span> <span class="o">=</span> <span class="n">filter_shape</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">image_shape</span> <span class="o">=</span> <span class="n">image_shape</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">poolsize</span> <span class="o">=</span> <span class="n">poolsize</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="o">=</span><span class="n">activation_fn</span>
        <span class="c1"># initialize weights and biases</span>
        <span class="n">n_out</span> <span class="o">=</span> <span class="p">(</span><span class="n">filter_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">(</span><span class="n">filter_shape</span><span class="p">[</span><span class="mi">2</span><span class="p">:])</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">prod</span><span class="p">(</span><span class="n">poolsize</span><span class="p">))</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">w</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span>
                <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n_out</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="n">filter_shape</span><span class="p">),</span>
                <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span>
                <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">filter_shape</span><span class="p">[</span><span class="mi">0</span><span class="p">],)),</span>
                <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">]</span>

    <span class="k">def</span> <span class="nf">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inpt</span><span class="p">,</span> <span class="n">inpt_dropout</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt</span> <span class="o">=</span> <span class="n">inpt</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">image_shape</span><span class="p">)</span>
        <span class="n">conv_out</span> <span class="o">=</span> <span class="n">conv</span><span class="o">.</span><span class="n">conv2d</span><span class="p">(</span>
            <span class="nb">input</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt</span><span class="p">,</span> <span class="n">filters</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="n">filter_shape</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">filter_shape</span><span class="p">,</span>
            <span class="n">image_shape</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">image_shape</span><span class="p">)</span>
        <span class="n">pooled_out</span> <span class="o">=</span> <span class="n">downsample</span><span class="o">.</span><span class="n">max_pool_2d</span><span class="p">(</span>
            <span class="nb">input</span><span class="o">=</span><span class="n">conv_out</span><span class="p">,</span> <span class="n">ds</span><span class="o">=</span><span class="bp">self</span><span class="o">.</span><span class="n">poolsize</span><span class="p">,</span> <span class="n">ignore_border</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="p">(</span>
            <span class="n">pooled_out</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="o">.</span><span class="n">dimshuffle</span><span class="p">(</span><span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="s1">&#39;x&#39;</span><span class="p">,</span> <span class="s1">&#39;x&#39;</span><span class="p">))</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="c1"># no dropout in the convolutional layers</span>

<span class="k">class</span> <span class="nc">FullyConnectedLayer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>

    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">,</span> <span class="n">activation_fn</span><span class="o">=</span><span class="n">sigmoid</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span> <span class="o">=</span> <span class="n">n_in</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_out</span> <span class="o">=</span> <span class="n">n_out</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span> <span class="o">=</span> <span class="n">activation_fn</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span> <span class="o">=</span> <span class="n">p_dropout</span>
        <span class="c1"># Initialize weights and biases</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">w</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span>
                <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span>
                    <span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mf">1.0</span><span class="o">/</span><span class="n">n_out</span><span class="p">),</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">)),</span>
                <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">asarray</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">scale</span><span class="o">=</span><span class="mf">1.0</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="n">n_out</span><span class="p">,)),</span>
                       <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">]</span>

    <span class="k">def</span> <span class="nf">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inpt</span><span class="p">,</span> <span class="n">inpt_dropout</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt</span> <span class="o">=</span> <span class="n">inpt</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">))</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="p">(</span>
            <span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span><span class="o">*</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span> <span class="o">=</span> <span class="n">dropout_layer</span><span class="p">(</span>
            <span class="n">inpt_dropout</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">activation_fn</span><span class="p">(</span>
            <span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>

    <span class="k">def</span> <span class="nf">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
        <span class="s2">&quot;Return the accuracy for the mini-batch.&quot;</span>
        <span class="k">return</span> <span class="n">T</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span><span class="p">))</span>

<span class="k">class</span> <span class="nc">SoftmaxLayer</span><span class="p">(</span><span class="nb">object</span><span class="p">):</span>

    <span class="k">def</span> <span class="nf">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">,</span> <span class="n">p_dropout</span><span class="o">=</span><span class="mf">0.0</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span> <span class="o">=</span> <span class="n">n_in</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">n_out</span> <span class="o">=</span> <span class="n">n_out</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span> <span class="o">=</span> <span class="n">p_dropout</span>
        <span class="c1"># Initialize weights and biases</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">w</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_in</span><span class="p">,</span> <span class="n">n_out</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;w&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">b</span> <span class="o">=</span> <span class="n">theano</span><span class="o">.</span><span class="n">shared</span><span class="p">(</span>
            <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_out</span><span class="p">,),</span> <span class="n">dtype</span><span class="o">=</span><span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">),</span>
            <span class="n">name</span><span class="o">=</span><span class="s1">&#39;b&#39;</span><span class="p">,</span> <span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">params</span> <span class="o">=</span> <span class="p">[</span><span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">]</span>

    <span class="k">def</span> <span class="nf">set_inpt</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">inpt</span><span class="p">,</span> <span class="n">inpt_dropout</span><span class="p">,</span> <span class="n">mini_batch_size</span><span class="p">):</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt</span> <span class="o">=</span> <span class="n">inpt</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">))</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">((</span><span class="mi">1</span><span class="o">-</span><span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span><span class="o">*</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span> <span class="o">=</span> <span class="n">T</span><span class="o">.</span><span class="n">argmax</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">output</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span> <span class="o">=</span> <span class="n">dropout_layer</span><span class="p">(</span>
            <span class="n">inpt_dropout</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">mini_batch_size</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_in</span><span class="p">)),</span> <span class="bp">self</span><span class="o">.</span><span class="n">p_dropout</span><span class="p">)</span>
        <span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span> <span class="o">=</span> <span class="n">softmax</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">inpt_dropout</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">w</span><span class="p">)</span> <span class="o">+</span> <span class="bp">self</span><span class="o">.</span><span class="n">b</span><span class="p">)</span>

    <span class="k">def</span> <span class="nf">cost</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">net</span><span class="p">):</span>
        <span class="s2">&quot;Return the log-likelihood cost.&quot;</span>
        <span class="k">return</span> <span class="o">-</span><span class="n">T</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">output_dropout</span><span class="p">)[</span><span class="n">T</span><span class="o">.</span><span class="n">arange</span><span class="p">(</span><span class="n">net</span><span class="o">.</span><span class="n">y</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]),</span> <span class="n">net</span><span class="o">.</span><span class="n">y</span><span class="p">])</span>

    <span class="k">def</span> <span class="nf">accuracy</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">y</span><span class="p">):</span>
        <span class="s2">&quot;Return the accuracy for the mini-batch.&quot;</span>
        <span class="k">return</span> <span class="n">T</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">T</span><span class="o">.</span><span class="n">eq</span><span class="p">(</span><span class="n">y</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">y_out</span><span class="p">))</span>


<span class="c1">#### Miscellanea</span>
<span class="k">def</span> <span class="nf">size</span><span class="p">(</span><span class="n">data</span><span class="p">):</span>
    <span class="s2">&quot;Return the size of the dataset `data`.&quot;</span>
    <span class="k">return</span> <span class="n">data</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">get_value</span><span class="p">(</span><span class="n">borrow</span><span class="o">=</span><span class="bp">True</span><span class="p">)</span><span class="o">.</span><span class="n">shape</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span>

<span class="k">def</span> <span class="nf">dropout_layer</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="n">p_dropout</span><span class="p">):</span>
    <span class="n">srng</span> <span class="o">=</span> <span class="n">shared_randomstreams</span><span class="o">.</span><span class="n">RandomStreams</span><span class="p">(</span>
        <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">RandomState</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span><span class="o">.</span><span class="n">randint</span><span class="p">(</span><span class="mi">999999</span><span class="p">))</span>
    <span class="n">mask</span> <span class="o">=</span> <span class="n">srng</span><span class="o">.</span><span class="n">binomial</span><span class="p">(</span><span class="n">n</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span> <span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="o">-</span><span class="n">p_dropout</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="n">layer</span><span class="o">.</span><span class="n">shape</span><span class="p">)</span>
    <span class="k">return</span> <span class="n">layer</span><span class="o">*</span><span class="n">T</span><span class="o">.</span><span class="n">cast</span><span class="p">(</span><span class="n">mask</span><span class="p">,</span> <span class="n">theano</span><span class="o">.</span><span class="n">config</span><span class="o">.</span><span class="n">floatX</span><span class="p">)</span>
</pre>
    </div>
    </p>
    <p>
    <h4><a name="problems_269956"></a><a href="#problems_269956">Problems</a></h4>
    <ul>
      </p>
      <p>
        <li> At present, the <tt>SGD</tt> method requires the user to manually
          choose the number of epochs to train for. Earlier in the book we
          discussed an automated way of selecting the number of epochs to
          train for, known as <a href="chap3.html#early_stopping">early
            stopping</a>. Modify <tt>network3.py</tt> to implement early stopping.
      </p>
      <p>
        <li> Add a <tt>Network</tt> method to return the accuracy on an
          arbitrary data set.
      </p>
      <p>
        <li> Modify the <tt>SGD</tt> method to allow the learning rate $\eta$
          to be a function of the epoch number. <em>Hint: After working on
            this problem for a while, you may find it useful to see the
            discussion at
            <a href="https://groups.google.com/forum/#!topic/theano-users/NQ9NYLvleGc">this
              link</a>.</em>
      </p>
      <p>
        <li> Earlier in the chapter I described a technique for expanding the
          training data by applying (small) rotations, skewing, and
          translation. Modify <tt>network3.py</tt> to incorporate all these
          techniques. <em>Note: Unless you have a tremendous amount of
            memory, it is not practical to explicitly generate the entire
            expanded data set. So you should consider alternate approaches.</em>
      </p>
      <p>
        <li> Add the ability to load and save networks to <tt>network3.py</tt>.
      </p>
      <p>
        <li> A shortcoming of the current code is that it provides few
          diagnostic tools. Can you think of any diagnostics to add that
          would make it easier to understand to what extent a network is
          overfitting? Add them.
      </p>
      <p>
        <li> We've used the same initialization procedure for rectified
          linear units as for sigmoid (and tanh) neurons. Our
          <a href="chap3.html#weight_initialization">argument for that
            initialization</a> was specific to the sigmoid function. Consider a
          network made entirely of rectified linear units (including outputs).
          Show that rescaling all the weights in the network by a constant
          factor $c > 0$ simply rescales the outputs by a factor $c^{L-1}$,
          where $L$ is the number of layers. How does this change if the
          final layer is a softmax? What do you think of using the sigmoid
          initialization procedure for the rectified linear units? Can you
          think of a better initialization procedure? <em>Note: This is a
            very open-ended problem, not something with a simple
            self-contained answer. Still, considering the problem will help
            you better understand networks containing rectified linear units.</em>
      </p>
      <p>
        <li> Our
          <a
            href="chap5.html#what's_causing_the_vanishing_gradient_problem_unstable_gradients_in_deep_neural_nets">analysis</a>
          of the unstable gradient problem was for sigmoid neurons. How does
          the analysis change for networks made up of rectified linear units?
          Can you think of a good way of modifying such a network so it
          doesn't suffer from the unstable gradient problem? <em>Note: The
            word good in the second part of this makes the problem a research
            problem. It's actually easy to think of ways of making such
            modifications. But I haven't investigated in enough depth to know
            of a really good technique.</em>
    </ul>
    </p>
    <p>
    <h3><a name="recent_progress_in_image_recognition"></a><a href="#recent_progress_in_image_recognition">Recent
        progress in image recognition</a></h3>
    </p>
    <p>
      In 1998, the year MNIST was introduced, it took weeks to train a
      state-of-the-art workstation to achieve accuracies substantially worse
      than those we can achieve using a GPU and less than an hour of
      training. Thus, MNIST is no longer a problem that pushes the limits of
      available technique; rather, the speed of training means that it is a
      problem good for teaching and learning purposes. Meanwhile, the focus
      of research has moved on, and modern work involves much more
      challenging image recognition problems. In this section, I briefly
      describe some recent work on image recognition using neural networks.</p>
    <p>The section is different to most of the book. Through the book I've
      focused on ideas likely to be of lasting interest - ideas such as
      backpropagation, regularization, and convolutional networks. I've
      tried to avoid results which are fashionable as I write, but whose
      long-term value is unknown. In science, such results are more often
      than not ephemera which fade and have little lasting impact. Given
      this, a skeptic might say: "well, surely the recent progress in image
      recognition is an example of such ephemera? In another two or three
      years, things will have moved on. So surely these results are only of
      interest to a few specialists who want to compete at the absolute
      frontier? Why bother discussing it?"</p>
    <p>Such a skeptic is right that some of the finer details of recent
      papers will gradually diminish in perceived importance. With that
      said, the past few years have seen extraordinary improvements using
      deep nets to attack extremely difficult image recognition tasks.
      Imagine a historian of science writing about computer vision in the
      year 2100. They will identify the years 2011 to 2015 (and probably a
      few years beyond) as a time of huge breakthroughs, driven by deep
      convolutional nets. That doesn't mean deep convolutional nets will
      still be used in 2100, much less detailed ideas such as dropout,
      rectified linear units, and so on. But it does mean that an important
      transition is taking place, right now, in the history of ideas. It's
      a bit like watching the discovery of the atom, or the invention of
      antibiotics: invention and discovery on a historic scale. And so
      while we won't dig down deep into details, it's worth getting some
      idea of the exciting discoveries currently being made.</p>
    <p><strong>The 2012 LRMD paper:</strong> Let me start with a 2012
      paper*<span class="marginnote">
        *<a href="http://research.google.com/pubs/pub38115.html">Building
          high-level features using large scale unsupervised learning</a>, by
        Quoc Le, Marc'Aurelio Ranzato, Rajat Monga, Matthieu Devin, Kai
        Chen, Greg Corrado, Jeff Dean, and Andrew Ng (2012). Note that the
        detailed architecture of the network used in the paper differed in
        many details from the deep convolutional networks we've been
        studying. Broadly speaking, however, LRMD is based on many similar
        ideas.</span> from a group of researchers from Stanford and Google. I'll
      refer to this paper as LRMD, after the last names of the first four
      authors. LRMD used a neural network to classify images from
      <a href="http://www.image-net.org">ImageNet</a>, a very challenging image
      recognition problem. The 2011 ImageNet data that they used included
      16 million full color images, in 20 thousand categories. The images
      were crawled from the open net, and classified by workers from
      Amazon's Mechanical Turk service. Here's a few ImageNet
      images*<span class="marginnote">
        *These are from the 2014 dataset, which is somewhat
        changed from 2011. Qualitatively, however, the dataset is extremely
        similar. Details about ImageNet are available in the original
        ImageNet paper,
        <a href="http://www.image-net.org/papers/imagenet_cvpr09.pdf">ImageNet:
          a large-scale hierarchical image database</a>, by Jia Deng, Wei Dong,
        Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei (2009).</span>:
    </p>
    <p><img src="images/imagenet1.jpg" height="120px"><img src="images/imagenet2.jpg" height="120px"><img
        src="images/imagenet3.jpg" height="120px"><img src="images/imagenet4.jpg" height="120px"></p>
    <p>These are, respectively, in the categories for beading plane, brown
      root rot fungus, scalded milk, and the common roundworm. If you're
      looking for a challenge, I encourage you to visit ImageNet's list of
      <a href="http://www.image-net.org/synset?wnid=n03489162">hand tools</a>,
      which distinguishes between beading planes, block planes, chamfer
      planes, and about a dozen other types of plane, amongst other
      categories. I don't know about you, but I cannot confidently
      distinguish between all these tool types. This is obviously a much
      more challenging image recognition task than MNIST! LRMD's network
      obtained a respectable $15.8$ percent accuracy for correctly
      classifying ImageNet images. That may not sound impressive, but it
      was a huge improvement over the previous best result of $9.3$ percent
      accuracy. That jump suggested that neural networks might offer a
      powerful approach to very challenging image recognition tasks, such as
      ImageNet.
    </p>
    <p><strong>The 2012 KSH paper:</strong> The work of LRMD was followed by a 2012
      paper of Krizhevsky, Sutskever and Hinton
      (KSH)*<span class="marginnote">
        *<a href="http://www.cs.toronto.edu/&#126;fritz/absps/imagenet.pdf">ImageNet
          classification with deep convolutional neural networks</a>, by Alex
        Krizhevsky, Ilya Sutskever, and Geoffrey E. Hinton (2012).</span>. KSH
      trained and tested a deep convolutional neural network using a
      restricted subset of the ImageNet data. The subset they used came from
      a popular machine learning competition - the ImageNet Large-Scale
      Visual Recognition Challenge (ILSVRC). Using a competition dataset
      gave them a good way of comparing their approach to other leading
      techniques. The ILSVRC-2012 training set contained about 1.2 million
      ImageNet images, drawn from 1,000 categories. The validation and test
      sets contained 50,000 and 150,000 images, respectively, drawn from the
      same 1,000 categories.</p>
    <p>One difficulty in running the ILSVRC competition is that many ImageNet
      images contain multiple objects. Suppose an image shows a labrador
      retriever chasing a soccer ball. The so-called "correct" ImageNet
      classification of the image might be as a labrador retriever. Should
      an algorithm be penalized if it labels the image as a soccer ball?
      Because of this ambiguity, an algorithm was considered correct if the
      actual ImageNet classification was among the $5$ classifications the
      algorithm considered most likely. By this top-$5$ criterion, KSH's
      deep convolutional network achieved an accuracy of $84.7$ percent,
      vastly better than the next-best contest entry, which achieved an
      accuracy of $73.8$ percent. Using the more restrictive metric of
      getting the label exactly right, KSH's network achieved an accuracy of
      $63.3$ percent.</p>
    <p>It's worth briefly describing KSH's network, since it has inspired
      much subsequent work. It's also, as we shall see, closely related to
      the networks we trained earlier in this chapter, albeit more
      elaborate. KSH used a deep convolutional neural network, trained on
      two GPUs. They used two GPUs because the particular type of GPU they
      were using (an NVIDIA GeForce GTX 580) didn't have enough on-chip
      memory to store their entire network. So they split the network into
      two parts, partitioned across the two GPUs.</p>
    <p>The KSH network has $7$ layers of hidden neurons. The first $5$
      hidden layers are convolutional layers (some with max-pooling), while
      the next $2$ layers are fully-connected layers. The output layer is a
      $1,000$-unit softmax layer, corresponding to the $1,000$ image
      classes. Here's a sketch of the network, taken from the KSH
      paper*<span class="marginnote">
        *Thanks to Ilya Sutskever.</span>. The details are explained
      below. Note that many layers are split into $2$ parts, corresponding
      to the $2$ GPUs.</p>
    <p><img src="images/KSH.jpg" width="600px"></p>
    <p>The input layer contains $3 \times 224 \times 224$ neurons,
      representing the RGB values for a $224 \times 224$ image. Recall
      that, as mentioned earlier, ImageNet contains images of varying
      resolution. This poses a problem, since a neural network's input
      layer is usually of a fixed size. KSH dealt with this by rescaling
      each image so the shorter side had length $256$. They then cropped out
      a $256 \times 256$ area in the center of the rescaled image. Finally,
      KSH extracted random $224 \times 224$ subimages (and horizontal
      reflections) from the $256 \times 256$ images. They did this random
      cropping as a way of expanding the training data, and thus reducing
      overfitting. This is particularly helpful in a large network such as
      KSH's. It was these $224 \times 224$ images which were used as inputs
      to the network. In most cases the cropped image still contains the
      main object from the uncropped image.</p>
    <p>Moving on to the hidden layers in KSH's network, the first hidden
      layer is a convolutional layer, with a max-pooling step. It uses
      local receptive fields of size $11 \times 11$, and a stride length of
      $4$ pixels. There are a total of $96$ feature maps. The feature maps
      are split into two groups of $48$ each, with the first $48$ feature
      maps residing on one GPU, and the second $48$ feature maps residing on
      the other GPU. The max-pooling in this and later layers is done in $3
      \times 3$ regions, but the pooling regions are allowed to overlap, and
      are just $2$ pixels apart.</p>
    <p>The second hidden layer is also a convolutional layer, with a
      max-pooling step. It uses $5 \times 5$ local receptive fields, and
      there's a total of $256$ feature maps, split into $128$ on each GPU.
      Note that the feature maps only use $48$ input channels, not the full
      $96$ output from the previous layer (as would usually be the case).
      This is because any single feature map only uses inputs from the same
      GPU. In this sense the network departs from the convolutional
      architecture we described earlier in the chapter, though obviously the
      basic idea is still the same.</p>
    <p>The third, fourth and fifth hidden layers are convolutional layers,
      but unlike the previous layers, they do not involve max-pooling.
      Their respectives parameters are: (3) $384$ feature maps, with $3
      \times 3$ local receptive fields, and $256$ input channels; (4) $384$
      feature maps, with $3 \times 3$ local receptive fields, and $192$
      input channels; and (5) $256$ feature maps, with $3 \times 3$ local
      receptive fields, and $192$ input channels. Note that the third layer
      involves some inter-GPU communication (as depicted in the figure) in
      order that the feature maps use all $256$ input channels.</p>
    <p>The sixth and seventh hidden layers are fully-connected layers, with
      $4,096$ neurons in each layer.</p>
    <p>The output layer is a $1,000$-unit softmax layer.</p>
    <p>The KSH network takes advantage of many techniques. Instead of using
      the sigmoid or tanh activation functions, KSH use rectified linear
      units, which sped up training significantly. KSH's network had
      roughly 60 million learned parameters, and was thus, even with the
      large training set, susceptible to overfitting. To overcome this,
      they expanded the training set using the random cropping strategy we
      discussed above. They also further addressed overfitting by using a
      variant of <a href="chap3.html#regularization">l2 regularization</a>, and
      <a href="chap3.html#other_techniques_for_regularization">dropout</a>.
      The network itself was trained using
      <a href="chap3.html#variations_on_stochastic_gradient_descent">momentum-based</a>
      mini-batch stochastic gradient descent.
    </p>
    <p>That's an overview of many of the core ideas in the KSH paper. I've
      omitted some details, for which you should look at the paper. You can
      also look at Alex Krizhevsky's
      <a href="https://code.google.com/p/cuda-convnet/">cuda-convnet</a> (and
      successors), which contains code implementing many of the ideas. A
      Theano-based implementation has also been
      developed*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1412.2302">Theano-based
          large-scale visual recognition with multiple GPUs</a>, by Weiguang
        Ding, Ruoyan Wang, Fei Mao, and Graham Taylor (2014).</span>, with the
      code available
      <a href="https://github.com/uoguelph-mlrg/theano_alexnet">here</a>. The
      code is recognizably along similar lines to that developed in this
      chapter, although the use of multiple GPUs complicates things
      somewhat. The Caffe neural nets framework also includes a version of
      the KSH network, see their
      <a href="http://caffe.berkeleyvision.org/model_zoo.html">Model Zoo</a> for
      details.
    </p>
    <p><strong>The 2014 ILSVRC competition:</strong> Since 2012, rapid progress
      continues to be made. Consider the 2014 ILSVRC competition. As in
      2012, it involved a training set of $1.2$ million images, in $1,000$
      categories, and the figure of merit was whether the top $5$
      predictions included the correct category. The winning team, based
      primarily at
      Google*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1409.4842">Going deeper
          with convolutions</a>, by Christian Szegedy, Wei Liu, Yangqing Jia,
        Pierre Sermanet, Scott Reed, Dragomir Anguelov, Dumitru Erhan,
        Vincent Vanhoucke, and Andrew Rabinovich (2014).</span>, used a deep
      convolutional network with $22$ layers of neurons. They called their
      network GoogLeNet, as a homage to LeNet-5. GoogLeNet achieved a top-5
      accuracy of $93.33$ percent, a giant improvement over the 2013 winner
      (<a href="http://www.clarifai.com">Clarifai</a>, with $88.3$ percent), and
      the 2012 winner (KSH, with $84.7$ percent).</p>
    <p>Just how good is GoogLeNet's $93.33$ percent accuracy? In 2014 a team
      of researchers wrote a survey paper about the ILSVRC
      competition*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1409.0575">ImageNet
          large scale visual recognition challenge</a>, by Olga Russakovsky,
        Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma,
        Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein,
        Alexander C. Berg, and Li Fei-Fei (2014).</span>. One of the questions
      they address is how well humans perform on ILSVRC. To do this, they
      built a system which lets humans classify ILSVRC images. As one of
      the authors, Andrej Karpathy, explains in an informative
      <a href="http://karpathy.github.io/2014/09/02/what-i-learned-from-competing-against-a-convnet-on-imagenet/">blog
        post</a>, it was a lot of trouble to get the humans up to GoogLeNet's
      performance:
    </p>
    <p>
    <blockquote> ...the task of labeling images with 5 out of 1000
      categories quickly turned out to be extremely challenging, even for
      some friends in the lab who have been working on ILSVRC and its
      classes for a while. First we thought we would put it up on [Amazon
      Mechanical Turk]. Then we thought we could recruit paid
      undergrads. Then I organized a labeling party of intense labeling
      effort only among the (expert labelers) in our lab. Then I developed
      a modified interface that used GoogLeNet predictions to prune the
      number of categories from 1000 to only about 100. It was still too
      hard - people kept missing categories and getting up to ranges of
      13-15&#37; error rates. In the end I realized that to get anywhere
      competitively close to GoogLeNet, it was most efficient if I sat
      down and went through the painfully long training process and the
      subsequent careful annotation process myself... The labeling
      happened at a rate of about 1 per minute, but this decreased over
      time... Some images are easily recognized, while some images (such
      as those of fine-grained breeds of dogs, birds, or monkeys) can
      require multiple minutes of concentrated effort. I became very good
      at identifying breeds of dogs... Based on the sample of images I
      worked on, the GoogLeNet classification error turned out to be
      6.8&#37;... My own error in the end turned out to be 5.1&#37;,
      approximately 1.7&#37; better. </blockquote>
    </p>
    <p>In other words, an expert human, working painstakingly, was with great
      effort able to narrowly beat the deep neural network. In fact,
      Karpathy reports that a second human expert, trained on a smaller
      sample of images, was only able to attain a $12.0$ percent top-5 error
      rate, significantly below GoogLeNet's performance. About half the
      errors were due to the expert "failing to spot and consider the
      ground truth label as an option".</p>
    <p>These are astonishing results. Indeed, since this work, several teams
      have reported systems whose top-5 error rate is actually <em>better</em>
      than 5.1&#37;. This has sometimes been reported in the media as the
      systems having better-than-human vision. While the results are
      genuinely exciting, there are many caveats that make it misleading to
      think of the systems as having better-than-human vision. The ILSVRC
      challenge is in many ways a rather limited problem - a crawl of the
      open web is not necessarily representative of images found in
      applications! And, of course, the top-$5$ criterion is quite
      artificial. We are still a long way from solving the problem of image
      recognition or, more broadly, computer vision. Still, it's extremely
      encouraging to see so much progress made on such a challenging
      problem, over just a few years.</p>
    <p><strong>Other activity:</strong> I've focused on ImageNet, but there's a
      considerable amount of other activity using neural nets to do image
      recognition. Let me briefly describe a few interesting recent
      results, just to give the flavour of some current work.</p>
    <p>One encouraging practical set of results comes from a team at Google,
      who applied deep convolutional networks to the problem of recognizing
      street numbers in Google's Street View
      imagery*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1312.6082">Multi-digit
          Number Recognition from Street View Imagery using Deep
          Convolutional Neural Networks</a>, by Ian J. Goodfellow, Yaroslav
        Bulatov, Julian Ibarz, Sacha Arnoud, and Vinay Shet (2013).</span>. In
      their paper, they report detecting and automatically transcribing
      nearly 100 million street numbers at an accuracy similar to that of a
      human operator. The system is fast: their system transcribed all of
      Street View's images of street numbers in France in less than an hour!
      They say: "Having this new dataset significantly increased the
      geocoding quality of Google Maps in several countries especially the
      ones that did not already have other sources of good geocoding." And
      they go on to make the broader claim: "We believe with this model we
      have solved [optical character recognition] for short sequences [of
      characters] for many applications."</p>
    <p>I've perhaps given the impression that it's all a parade of
      encouraging results. Of course, some of the most interesting work
      reports on fundamental things we don't yet understand. For instance,
      a 2013 paper*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1312.6199">Intriguing
          properties of neural networks</a>, by Christian Szegedy, Wojciech
        Zaremba, Ilya Sutskever, Joan Bruna, Dumitru Erhan, Ian Goodfellow,
        and Rob Fergus (2013)</span> showed that deep networks may suffer from
      what are effectively blind spots. Consider the lines of images below.
      On the left is an ImageNet image classified correctly by their
      network. On the right is a slightly perturbed image (the perturbation
      is in the middle) which is classified <em>incorrectly</em> by the
      network. The authors found that there are such "adversarial" images
      for every sample image, not just a few special ones.</p>
    <p><img src="images/adversarial.jpg"></p>
    <p>This is a disturbing result. The paper used a network based on the
      same code as KSH's network - that is, just the type of network that
      is being increasingly widely used. While such neural networks compute
      functions which are, in principle, continuous, results like this
      suggest that in practice they're likely to compute functions which are
      very nearly discontinuous. Worse, they'll be discontinuous in ways
      that violate our intuition about what is reasonable behavior. That's
      concerning. Furthermore, it's not yet well understood what's causing
      the discontinuity: is it something about the loss function? The
      activation functions used? The architecture of the network?
      Something else? We don't yet know.</p>
    <p>Now, these results are not quite as bad as they sound. Although such
      adversarial images are common, they're also unlikely in practice. As
      the paper notes:</p>
    <p>
    <blockquote>
      The existence of the adversarial negatives appears to be in
      contradiction with the network’s ability to achieve high
      generalization performance. Indeed, if the network can generalize
      well, how can it be confused by these adversarial negatives, which
      are indistinguishable from the regular examples? The explanation is
      that the set of adversarial negatives is of extremely low
      probability, and thus is never (or rarely) observed in the test set,
      yet it is dense (much like the rational numbers), and so it is found
      near virtually every test case.
    </blockquote>
    </p>
    <p>Nonetheless, it is distressing that we understand neural nets so
      poorly that this kind of result should be a recent discovery. Of
      course, a major benefit of the results is that they have stimulated
      much followup work. For example, one recent
      paper*<span class="marginnote">
        *<a href="http://arxiv.org/abs/1412.1897">Deep Neural
          Networks are Easily Fooled: High Confidence Predictions for
          Unrecognizable Images</a>, by Anh Nguyen, Jason Yosinski, and Jeff
        Clune (2014).</span> shows that given a trained network it's possible to
      generate images which look to a human like white noise, but which the
      network classifies as being in a known category with a very high
      degree of confidence. This is another demonstration that we have a
      long way to go in understanding neural networks and their use in image
      recognition.</p>
    <p>Despite results like this, the overall picture is encouraging. We're
      seeing rapid progress on extremely difficult benchmarks, like
      ImageNet. We're also seeing rapid progress in the solution of
      real-world problems, like recognizing street numbers in StreetView.
      But while this is encouraging it's not enough just to see improvements
      on benchmarks, or even real-world applications. There are fundamental
      phenomena which we still understand poorly, such as the existence of
      adversarial images. When such fundamental problems are still being
      discovered (never mind solved), it is premature to say that we're near
      solving the problem of image recognition. At the same time such
      problems are an exciting stimulus to further work.</p>
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    <p>
    <h3><a name="other_approaches_to_deep_neural_nets"></a><a href="#other_approaches_to_deep_neural_nets">Other
        approaches to deep neural nets</a></h3>
    </p>
    <p>Through this book, we've concentrated on a single problem: classifying
      the MNIST digits. It's a juicy problem which forced us to understand
      many powerful ideas: stochastic gradient descent, backpropagation,
      convolutional nets, regularization, and more. But it's also a narrow
      problem. If you read the neural networks literature, you'll run into
      many ideas we haven't discussed: recurrent neural networks, Boltzmann
      machines, generative models, transfer learning, reinforcement
      learning, and so on, on and on $\ldots$ and on! Neural networks is a
      vast field. However, many important ideas are variations on ideas
      we've already discussed, and can be understood with a little effort.
      In this section I provide a glimpse of these as yet unseen vistas.
      The discussion isn't detailed, nor comprehensive - that would
      greatly expand the book. Rather, it's impressionistic, an attempt to
      evoke the conceptual richness of the field, and to relate some of
      those riches to what we've already seen. Through the section, I'll
      provide a few links to other sources, as entrees to learn more. Of
      course, many of these links will soon be superseded, and you may wish
      to search out more recent literature. That point notwithstanding, I
      expect many of the underlying ideas to be of lasting interest.</p>
    <p><strong>Recurrent neural networks (RNNs):</strong> In the feedforward nets
      we've been using there is a single input which completely determines
      the activations of all the neurons through the remaining layers. It's
      a very static picture: everything in the network is fixed, with a
      frozen, crystalline quality to it. But suppose we allow the elements
      in the network to keep changing in a dynamic way. For instance, the
      behaviour of hidden neurons might not just be determined by the
      activations in previous hidden layers, but also by the activations at
      earlier times. Indeed, a neuron's activation might be determined in
      part by its own activation at an earlier time. That's certainly not
      what happens in a feedforward network. Or perhaps the activations of
      hidden and output neurons won't be determined just by the current
      input to the network, but also by earlier inputs.</p>
    <p>Neural networks with this kind of time-varying behaviour are known as
      <em>recurrent neural networks</em> or <em>RNNs</em>. There are many
      different ways of mathematically formalizing the informal description
      of recurrent nets given in the last paragraph. You can get the
      flavour of some of these mathematical models by glancing at
      <a href="http://en.wikipedia.org/wiki/Recurrent_neural_network">the
        Wikipedia article on RNNs</a>. As I write, that page lists no fewer
      than 13 different models. But mathematical details aside, the broad
      idea is that RNNs are neural networks in which there is some notion of
      dynamic change over time. And, not surprisingly, they're particularly
      useful in analysing data or processes that change over time. Such
      data and processes arise naturally in problems such as speech or
      natural language, for example.
    </p>
    <p>One way RNNs are currently being used is to connect neural networks
      more closely to traditional ways of thinking about algorithms, ways of
      thinking based on concepts such as Turing machines and (conventional)
      programming languages. <a href="http://arxiv.org/abs/1410.4615">A 2014
        paper</a> developed an RNN which could take as input a
      character-by-character description of a (very, very simple!) Python
      program, and use that description to predict the output. Informally,
      the network is learning to "understand" certain Python programs.
      <a href="http://arxiv.org/abs/1410.5401">A second paper, also from 2014</a>,
      used RNNs as a starting point to develop what they called a neural
      Turing machine (NTM). This is a universal computer whose entire
      structure can be trained using gradient descent. They trained their
      NTM to infer algorithms for several simple problems, such as sorting
      and copying.
    </p>
    <p>As it stands, these are extremely simple toy models. Learning to
      execute the Python program <tt>print(398345+42598)</tt> doesn't make a
      network into a full-fledged Python interpreter! It's not clear how
      much further it will be possible to push the ideas. Still, the
      results are intriguing. Historically, neural networks have done well
      at pattern recognition problems where conventional algorithmic
      approaches have trouble. Vice versa, conventional algorithmic
      approaches are good at solving problems that neural nets aren't so
      good at. No-one today implements a web server or a database program
      using a neural network! It'd be great to develop unified models that
      integrate the strengths of both neural networks and more traditional
      approaches to algorithms. RNNs and ideas inspired by RNNs may help us
      do that.</p>
    <p>RNNs have also been used in recent years to attack many other
      problems. They've been particularly useful in speech recognition.
      Approaches based on RNNs have, for example,
      <a href="http://arxiv.org/abs/1303.5778">set records for the accuracy of
        phoneme recognition</a>. They've also been used to develop
      <a href="http://www.fit.vutbr.cz/&#126;imikolov/rnnlm/thesis.pdf">improved
        models of the language people use while speaking</a>. Better language
      models help disambiguate utterances that otherwise sound alike. A
      good language model will, for example, tell us that "to infinity and
      beyond" is much more likely than "two infinity and beyond", despite
      the fact that the phrases sound identical. RNNs have been used to set
      new records for certain language benchmarks.
    </p>
    <p>This work is, incidentally, part of a broader use of deep neural nets
      of all types, not just RNNs, in speech recognition. For example, an
      approach based on deep nets has achieved
      <a href="http://arxiv.org/abs/1309.1501">outstanding results on large
        vocabulary continuous speech recognition</a>. And another system based
      on deep nets has been deployed in
      <a href="http://www.wired.com/2013/02/android-neural-network/">Google's
        Android operating system</a> (for related technical work, see
      <a href="http://research.google.com/pubs/VincentVanhoucke.html">Vincent
        Vanhoucke's 2012-2015 papers</a>).
    </p>
    <p>I've said a little about what RNNs can do, but not so much about how
      they work. It perhaps won't surprise you to learn that many of the
      ideas used in feedforward networks can also be used in RNNs. In
      particular, we can train RNNs using straightforward modifications to
      gradient descent and backpropagation. Many other ideas used in
      feedforward nets, ranging from regularization techniques to
      convolutions to the activation and cost functions used, are also
      useful in recurrent nets. And so many of the techniques we've
      developed in the book can be adapted for use with RNNs.</p>
    <p></p>
    <p></p>
    <p></p>
    <p><strong>Long short-term memory units (LSTMs):</strong> One challenge affecting
      RNNs is that early models turned out to be very difficult to train,
      harder even than deep feedforward networks. The reason is the
      unstable gradient problem discussed in <a href="chap5.html">Chapter 5</a>.
      Recall that the usual manifestation of this problem is that the
      gradient gets smaller and smaller as it is propagated back through
      layers. This makes learning in early layers extremely slow. The
      problem actually gets worse in RNNs, since gradients aren't just
      propagated backward through layers, they're propagated backward
      through time. If the network runs for a long time that can make the
      gradient extremely unstable and hard to learn from. Fortunately, it's
      possible to incorporate an idea known as long short-term memory units
      (LSTMs) into RNNs. The units were introduced by
      <a href="http://dx.doi.org/10.1162/neco.1997.9.8.1735">Hochreiter and
        Schmidhuber in 1997</a> with the explicit purpose of helping address
      the unstable gradient problem. LSTMs make it much easier to get good
      results when training RNNs, and many recent papers (including many
      that I linked above) make use of LSTMs or related ideas.
    </p>
    <p><strong>Deep belief nets, generative models, and Boltzmann machines:</strong>
      Modern interest in deep learning began in 2006, with papers explaining
      how to train a type of neural network known as a <em>deep belief
        network</em> (DBN)*<span class="marginnote">
        *See
        <a href="http://www.cs.toronto.edu/&#126;hinton/absps/fastnc.pdf">A fast
          learning algorithm for deep belief nets</a>, by Geoffrey Hinton,
        Simon Osindero, and Yee-Whye Teh (2006), as well as the related work
        in
        <a href="http://www.sciencemag.org/content/313/5786/504.short">Reducing
          the dimensionality of data with neural networks</a>, by Geoffrey
        Hinton and Ruslan Salakhutdinov (2006).</span>. DBNs were influential for
      several years, but have since lessened in popularity, while models
      such as feedforward networks and recurrent neural nets have become
      fashionable. Despite this, DBNs have several properties that make
      them interesting.</p>
    <p>One reason DBNs are interesting is that they're an example of what's
      called a <em>generative model</em>. In a feedforward network, we
      specify the input activations, and they determine the activations of
      the feature neurons later in the network. A generative model like a
      DBN can be used in a similar way, but it's also possible to specify
      the values of some of the feature neurons and then "run the network
      backward", generating values for the input activations. More
      concretely, a DBN trained on images of handwritten digits can
      (potentially, and with some care) also be used to generate images that
      look like handwritten digits. In other words, the DBN would in some
      sense be learning to write. In this, a generative model is much like
      the human brain: not only can it read digits, it can also write them.
      In Geoffrey Hinton's memorable phrase,
      <a href="http://www.sciencedirect.com/science/article/pii/S0079612306650346">to
        recognize shapes, first learn to generate images</a>.
    </p>
    <p>A second reason DBNs are interesting is that they can do unsupervised
      and semi-supervised learning. For instance, when trained with image
      data, DBNs can learn useful features for understanding other images,
      even if the training images are unlabelled. And the ability to do
      unsupervised learning is extremely interesting both for fundamental
      scientific reasons, and - if it can be made to work well enough -
      for practical applications.</p>
    <p>Given these attractive features, why have DBNs lessened in popularity
      as models for deep learning? Part of the reason is that models such
      as feedforward and recurrent nets have achieved many spectacular
      results, such as their breakthroughs on image and speech recognition
      benchmarks. It's not surprising and quite right that there's now lots
      of attention being paid to these models. There's an unfortunate
      corollary, however. The marketplace of ideas often functions in a
      winner-take-all fashion, with nearly all attention going to the
      current fashion-of-the-moment in any given area. It can become
      extremely difficult for people to work on momentarily unfashionable
      ideas, even when those ideas are obviously of real long-term interest.
      My personal opinion is that DBNs and other generative models likely
      deserve more attention than they are currently receiving. And I won't
      be surprised if DBNs or a related model one day surpass the currently
      fashionable models. For an introduction to DBNs, see
      <a href="http://www.scholarpedia.org/article/Deep_belief_networks">this
        overview</a>. I've also found
      <a href="http://www.cs.toronto.edu/&#126;hinton/absps/guideTR.pdf">this
        article</a> helpful. It isn't primarily about deep belief nets,
      <em>per se</em>, but does contain much useful information about
      restricted Boltzmann machines, which are a key component of DBNs.
    </p>
    <p><strong>Other ideas:</strong> What else is going on in neural networks and
      deep learning? Well, there's a huge amount of other fascinating work.
      Active areas of research include using neural networks to do
      <a href="http://machinelearning.org/archive/icml2008/papers/391.pdf">natural
        language processing</a> (see <a href="http://arxiv.org/abs/1103.0398">also
        this informative review paper</a>),
      <a href="assets/MachineTranslation.pdf">machine translation</a>, as well as
      perhaps more surprising applications such as
      <a href="http://yann.lecun.com/exdb/publis/pdf/humphrey-jiis-13.pdf">music
        informatics</a>. There are, of course, many other areas too. In many
      cases, having read this book you should be able to begin following
      recent work, although (of course) you'll need to fill in gaps in
      presumed background knowledge.
    </p>
    <p>Let me finish this section by mentioning a particularly fun paper. It
      combines deep convolutional networks with a technique known as
      reinforcement learning in order to learn to
      <a href="http://www.cs.toronto.edu/&#126;vmnih/docs/dqn.pdf">play video games
        well</a> (see also
      <a href="http://www.nature.com/nature/journal/v518/n7540/abs/nature14236.html">this
        followup</a>). The idea is to use the convolutional network to
      simplify the pixel data from the game screen, turning it into a
      simpler set of features, which can be used to decide which action to
      take: "go left", "go down", "fire", and so on. What is
      particularly interesting is that a single network learned to play
      seven different classic video games pretty well, outperforming human
      experts on three of the games. Now, this all sounds like a stunt, and
      there's no doubt the paper was well marketed, with the title "Playing
      Atari with reinforcement learning". But looking past the surface
      gloss, consider that this system is taking raw pixel data - it
      doesn't even know the game rules! - and from that data learning to
      do high-quality decision-making in several very different and very
      adversarial environments, each with its own complex set of rules.
      That's pretty neat.
    </p>
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    <p>
    <h3><a name="on_the_future_of_neural_networks"></a><a href="#on_the_future_of_neural_networks">On the future of
        neural networks</a></h3>
    </p>
    <p><strong>Intention-driven user interfaces:</strong> There's an old joke in
      which an impatient professor tells a confused student: "don't listen
      to what I say; listen to what I <em>mean</em>". Historically,
      computers have often been, like the confused student, in the dark
      about what their users mean. But this is changing. I still remember
      my surprise the first time I misspelled a Google search query, only to
      have Google say "Did you mean [corrected query]?" and to offer the
      corresponding search results. Google CEO Larry Page
      <a href="http://googleblog.blogspot.ca/2012/08/building-search-engine-of-future-one.html">once
        described the perfect search engine as understanding exactly what
        [your queries] mean and giving you back exactly what you want</a>.
    </p>
    <p>This is a vision of an <em>intention-driven user interface</em>. In
      this vision, instead of responding to users' literal queries, search
      will use machine learning to take vague user input, discern precisely
      what was meant, and take action on the basis of those insights.</p>
    <p>The idea of intention-driven interfaces can be applied far more
      broadly than search. Over the next few decades, thousands of
      companies will build products which use machine learning to make user
      interfaces that can tolerate imprecision, while discerning and acting
      on the user's true intent. We're already seeing early examples of
      such intention-driven interfaces: Apple's Siri; Wolfram Alpha; IBM's
      Watson; systems which can
      <a href="http://arxiv.org/abs/1411.4555">annotate photos and videos</a>; and
      much more.
    </p>
    <p>Most of these products will fail. Inspired user interface design is
      hard, and I expect many companies will take powerful machine learning
      technology and use it to build insipid user interfaces. The best
      machine learning in the world won't help if your user interface
      concept stinks. But there will be a residue of products which
      succeed. Over time that will cause a profound change in how we relate
      to computers. Not so long ago - let's say, 2005 - users took it
      for granted that they needed precision in most interactions with
      computers. Indeed, computer literacy to a great extent meant
      internalizing the idea that computers are extremely literal; a single
      misplaced semi-colon may completely change the nature of an
      interaction with a computer. But over the next few decades I expect
      we'll develop many successful intention-driven user interfaces, and
      that will dramatically change what we expect when interacting with
      computers.</p>
    <p><strong>Machine learning, data science, and the virtuous circle of
        innovation:</strong> Of course, machine learning isn't just being used to
      build intention-driven interfaces. Another notable application is in
      data science, where machine learning is used to find the "known
      unknowns" hidden in data. This is already a fashionable area, and
      much has been written about it, so I won't say much. But I do want to
      mention one consequence of this fashion that is not so often remarked:
      over the long run it's possible the biggest breakthrough in machine
      learning won't be any single conceptual breakthrough. Rather, the
      biggest breakthrough will be that machine learning research becomes
      profitable, through applications to data science and other areas. If
      a company can invest 1 dollar in machine learning research and get 1
      dollar and 10 cents back reasonably rapidly, then a lot of money will
      end up in machine learning research. Put another way, machine
      learning is an engine driving the creation of several major new
      markets and areas of growth in technology. The result will be large
      teams of people with deep subject expertise, and with access to
      extraordinary resources. That will propel machine learning further
      forward, creating more markets and opportunities, a virtuous circle of
      innovation.</p>
    <p><strong>The role of neural networks and deep learning:</strong> I've been
      talking broadly about machine learning as a creator of new
      opportunities for technology. What will be the specific role of
      neural networks and deep learning in all this?</p>
    <p>To answer the question, it helps to look at history. Back in the
      1980s there was a great deal of excitement and optimism about neural
      networks, especially after backpropagation became widely known. That
      excitement faded, and in the 1990s the machine learning baton passed
      to other techniques, such as support vector machines. Today, neural
      networks are again riding high, setting all sorts of records,
      defeating all comers on many problems. But who is to say that
      tomorrow some new approach won't be developed that sweeps neural
      networks away again? Or perhaps progress with neural networks will
      stagnate, and nothing will immediately arise to take their place?</p>
    <p>For this reason, it's much easier to think broadly about the future of
      machine learning than about neural networks specifically. Part of the
      problem is that we understand neural networks so poorly. Why is it
      that neural networks can generalize so well? How is it that they
      avoid overfitting as well as they do, given the very large number of
      parameters they learn? Why is it that stochastic gradient descent
      works as well as it does? How well will neural networks perform as
      data sets are scaled? For instance, if ImageNet was expanded by a
      factor of $10$, would neural networks' performance improve more or
      less than other machine learning techniques? These are all simple,
      fundamental questions. And, at present, we understand the answers to
      these questions very poorly. While that's the case, it's difficult to
      say what role neural networks will play in the future of machine
      learning.</p>
    <p>I will make one prediction: I believe deep learning is here to stay.
      The ability to learn hierarchies of concepts, building up multiple
      layers of abstraction, seems to be fundamental to making sense of the
      world. This doesn't mean tomorrow's deep learners won't be radically
      different than today's. We could see major changes in the constituent
      units used, in the architectures, or in the learning algorithms.
      Those changes may be dramatic enough that we no longer think of the
      resulting systems as neural networks. But they'd still be doing deep
      learning.</p>
    <p><a name="AI"></a></p>
    <p><strong>Will neural networks and deep learning soon lead to artificial
        intelligence?</strong> In this book we've focused on using neural nets to
      do specific tasks, such as classifying images. Let's broaden our
      ambitions, and ask: what about general-purpose thinking computers?
      Can neural networks and deep learning help us solve the problem of
      (general) artificial intelligence (AI)? And, if so, given the rapid
      recent progress of deep learning, can we expect general AI any time
      soon?</p>
    <p>Addressing these questions comprehensively would take a separate book.
      Instead, let me offer one observation. It's based on an idea known as
      <a href="http://en.wikipedia.org/wiki/Conway%27s_law">Conway's law</a>:
    <blockquote>
      Any organization that designs a system... will inevitably produce a
      design whose structure is a copy of the organization's communication
      structure.
    </blockquote>
    So, for example, Conway's law suggests that the design of a Boeing 747
    aircraft will mirror the extended organizational structure of Boeing
    and its contractors at the time the 747 was designed. Or for a
    simple, specific example, consider a company building a complex
    software application. If the application's dashboard is supposed to
    be integrated with some machine learning algorithm, the person
    building the dashboard better be talking to the company's machine
    learning expert. Conway's law is merely that observation, writ large.</p>
    <p>Upon first hearing Conway's law, many people respond either "Well,
      isn't that banal and obvious?" or "Isn't that wrong?" Let me start
      with the objection that it's wrong. As an instance of this objection,
      consider the question: where does Boeing's accounting department show
      up in the design of the 747? What about their janitorial department?
      Their internal catering? And the answer is that these parts of the
      organization probably don't show up explicitly anywhere in the 747.
      So we should understand Conway's law as referring only to those parts
      of an organization concerned explicitly with design and engineering.</p>
    <p>What about the other objection, that Conway's law is banal and
      obvious? This may perhaps be true, but I don't think so, for
      organizations too often act with disregard for Conway's law. Teams
      building new products are often bloated with legacy hires or,
      contrariwise, lack a person with some crucial expertise. Think of all
      the products which have useless complicating features. Or think of
      all the products which have obvious major deficiencies - e.g., a
      terrible user interface. Problems in both classes are often caused by
      a mismatch between the team that was needed to produce a good product,
      and the team that was actually assembled. Conway's law may be
      obvious, but that doesn't mean people don't routinely ignore it.</p>
    <p>Conway's law applies to the design and engineering of systems where we
      start out with a pretty good understanding of the likely constituent
      parts, and how to build them. It can't be applied directly to the
      development of artificial intelligence, because AI isn't (yet) such a
      problem: we don't know what the constituent parts are. Indeed, we're
      not even sure what basic questions to be asking. In others words, at
      this point AI is more a problem of science than of engineering.
      Imagine beginning the design of the 747 without knowing about jet
      engines or the principles of aerodynamics. You wouldn't know what
      kinds of experts to hire into your organization. As Wernher von Braun
      put it, "basic research is what I'm doing when I don't know what I'm
      doing". Is there a version of Conway's law that applies to problems
      which are more science than engineering?</p>
    <p>To gain insight into this question, consider the history of medicine.
      In the early days, medicine was the domain of practitioners like Galen
      and Hippocrates, who studied the entire body. But as our knowledge
      grew, people were forced to specialize. We discovered many deep new
      ideas*<span class="marginnote">
        *My apologies for overloading "deep". I won't define
        "deep ideas" precisely, but loosely I mean the kind of idea which
        is the basis for a rich field of enquiry. The backpropagation
        algorithm and the germ theory of disease are both good examples.</span>:
      think of things like the germ theory of disease, for instance, or the
      understanding of how antibodies work, or the understanding that the
      heart, lungs, veins and arteries form a complete cardiovascular
      system. Such deep insights formed the basis for subfields such as
      epidemiology, immunology, and the cluster of inter-linked fields
      around the cardiovascular system. And so the structure of our
      knowledge has shaped the social structure of medicine. This is
      particularly striking in the case of immunology: realizing the immune
      system exists and is a system worthy of study is an extremely
      non-trivial insight. So we have an entire field of medicine - with
      specialists, conferences, even prizes, and so on - organized around
      something which is not just invisible, it's arguably not a distinct
      thing at all.</p>
    <p>This is a common pattern that has been repeated in many
      well-established sciences: not just medicine, but physics,
      mathematics, chemistry, and others. The fields start out monolithic,
      with just a few deep ideas. Early experts can master all those ideas.
      But as time passes that monolithic character changes. We discover
      many deep new ideas, too many for any one person to really master. As
      a result, the social structure of the field re-organizes and divides
      around those ideas. Instead of a monolith, we have fields within
      fields within fields, a complex, recursive, self-referential social
      structure, whose organization mirrors the connections between our
      deepest insights. <em>And so the structure of our knowledge shapes
        the social organization of science. But that social shape in turn
        constrains and helps determine what we can discover.</em> This is the
      scientific analogue of Conway's law.
    </p>
    <p></p>
    <p>So what's this got to do with deep learning or AI?</p>
    <p>Well, since the early days of AI there have been arguments about it
      that go, on one side, "Hey, it's not going to be so hard, we've got
      [super-special weapon] on our side", countered by "[super-special
      weapon] won't be enough". Deep learning is the latest super-special
      weapon I've heard used in such arguments*<span class="marginnote">
        *Interestingly, often
        not by leading experts in deep learning, who have been quite
        restrained. See, for example, this
        <a href="https://www.facebook.com/yann.lecun/posts/10152348155137143">thoughtful
          post</a> by Yann LeCun. This is a difference from many earlier
        incarnations of the argument.</span>; earlier versions of the argument
      used logic, or Prolog, or expert systems, or whatever the most
      powerful technique of the day was. The problem with such arguments is
      that they don't give you any good way of saying just how powerful any
      given candidate super-special weapon is. Of course, we've just spent
      a chapter reviewing evidence that deep learning can solve extremely
      challenging problems. It certainly looks very exciting and promising.
      But that was also true of systems like Prolog or
      <a href="http://en.wikipedia.org/wiki/Eurisko">Eurisko</a> or expert systems
      in their day. And so the mere fact that a set of ideas looks very
      promising doesn't mean much. How can we tell if deep learning is
      truly different from these earlier ideas? Is there some way of
      measuring how powerful and promising a set of ideas is? Conway's law
      suggests that as a rough and heuristic proxy metric we can evaluate
      the complexity of the social structure associated to those ideas.
    </p>
    <p>So, there are two questions to ask. First, how powerful a set of
      ideas are associated to deep learning, according to this metric of
      social complexity? Second, how powerful a theory will we need, in
      order to be able to build a general artificial intelligence?</p>
    <p>As to the first question: when we look at deep learning today, it's an
      exciting and fast-paced but also relatively monolithic field. There
      are a few deep ideas, and a few main conferences, with substantial
      overlap between several of the conferences. And there is paper after
      paper leveraging the same basic set of ideas: using stochastic
      gradient descent (or a close variation) to optimize a cost function.
      It's fantastic those ideas are so successful. But what we don't yet
      see is lots of well-developed subfields, each exploring their own sets
      of deep ideas, pushing deep learning in many directions. And so,
      according to the metric of social complexity, deep learning is, if
      you'll forgive the play on words, still a rather shallow field. It's
      still possible for one person to master most of the deepest ideas in
      the field.</p>
    <p>On the second question: how complex and powerful a set of ideas will
      be needed to obtain AI? Of course, the answer to this question is:
      no-one knows for sure. But in the <a href="sai.html">appendix</a> I examine
      some of the existing evidence on this question. I conclude that, even
      rather optimistically, it's going to take many, many deep ideas to
      build an AI. And so Conway's law suggests that to get to such a point
      we will necessarily see the emergence of many interrelating
      disciplines, with a complex and surprising structure mirroring the
      structure in our deepest insights. We don't yet see this rich social
      structure in the use of neural networks and deep learning. And so, I
      believe that we are several decades (at least) from using deep
      learning to develop general AI.</p>
    <p>I've gone to a lot of trouble to construct an argument which is
      tentative, perhaps seems rather obvious, and which has an indefinite
      conclusion. This will no doubt frustrate people who crave certainty.
      Reading around online, I see many people who loudly assert very
      definite, very strongly held opinions about AI, often on the basis of
      flimsy reasoning and non-existent evidence. My frank opinion is this:
      it's too early to say. As the old joke goes, if you ask a scientist
      how far away some discovery is and they say "10 years" (or more),
      what they mean is "I've got no idea". AI, like controlled fusion
      and a few other technologies, has been 10 years away for 60 plus
      years. On the flipside, what we definitely do have in deep learning
      is a powerful technique whose limits have not yet been found, and many
      wide-open fundamental problems. That's an exciting creative
      opportunity.</p>
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  <div class="footer"> <span class="left_footer"> In academic work,
      please cite this book as: Michael A. Nielsen, "Neural Networks and
      Deep Learning", Determination Press, 2015

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